Non-Traditional Stable Isotopes 9780939950980

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REVIEWS in MINERALOGY & GEOCHEMISTRY Volume 82

geochemical society

NON-TRADITIONAL STABLE ISOTOPES EDITORS: Fang-Zhen Teng, James M. Watkins and Nicolas Dauphas

MINERALOGICAL SOCIETY OF AMERICA GEOCHEMICAL SOCIETY Series Editor: Ian P. Swainson

2017

ISSN 1529-6466

REVIEWS IN MINERALOGY AND GEOCHEMISTRY Volume 82

2017

Non-Traditional Stable Isotopes EDITORS Fang-Zhen Teng University of Washington, USA

James Watkins University of Oregon, USA

Nicolas Dauphas The University of Chicago, USA

Front-cover: False-color image of a zoned olivine phenocryst (forsterite content) from the Kilauea Iki lava lake, Hawaii. The black marks are spots where the Fe isotopic composition of the olivine was measured by LA-MC-ICPMS and SIMS (Sio et al. 2013, Geochimica et Cosmochimica Acta 123, 302–321). In situ stable isotopic analyses of zoned minerals allow one to tell apart zoning produced by diffusion from zoning produced by growth from an evolving medium.

Series Editor: Ian Swainson MINERALOGICAL SOCIETY OF AMERICA GEOCHEMICAL SOCIETY

Reviews in Mineralogy and Geochemistry, Volume 82

Non-Traditional Stable Isotopes ISSN 1529-6466 ISBN 978-0-939950-98-0

Copyright 2017

The MINERALOGICAL SOCIETY of AMERICA 3635 Concorde Parkway, Suite 500 Chantilly, Virginia, 20151-1125, U.S.A. www.minsocam.org The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner’s consent that copies of the article can be made for personal use or internal use or for the personal use or internal use of specific clients, provided the original publication is cited. The consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other types of copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. For permission to reprint entire articles in these cases and the like, consult the Administrator of the Mineralogical Society of America as to the royalty due to the Society.

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FROM THE SERIES EDITOR It has been a pleasure working with the volume editors and authors on this 82nd volume of Reviews in Mineralogy and Geochemistry. Several chapters have associated supplemental figures and or tables that can be found at the MSA website. Any future errata will also be posted there. Ian P. Swainson, Series Editor Vienna, Austria

PREFACE Since the publication of Geochemistry of Non-Traditional Stable Isotopes in 2004 (volume 55 of Reviews in Mineralogy and Geochemistry), analytical techniques have significantly improved and new research directions have emerged in non-traditional stable isotope geochemistry. Our goal here is to review the current status of non-traditional isotope geochemistry from analytical, theoretical, and experimental approaches to analysis of natural samples. In particular, important applications to cosmochemistry, high-temperature geochemistry, low-temperature geochemistry, and geobiology are discussed. The aim of this volume is to provide the most comprehensive review on non-traditional isotope geochemistry for students and researchers who are interested in both the theory and applications of non-traditional stable isotope geochemistry. We take this opportunity to thank the timely contributions by authors of the individual chapters and insightful reviews from the following scientists: Bridget Bergquist, Greg Brennecka, Christopher Cloquet, Hans Eggenkamp, Toshiyuki Fujii, Sarah Gleeson, Doug Hammond, Adrianna Heimann, Richard Hervig, Fang Huang, Timm John, Tom Johnson, Abby Kavner, James Kubicki, Laura Lammers, Sheng-Ao Liu, Catherine Macris, Paul Mason, Ryan Mathur, Vasileios Mavromatis, Fred Moynier, Kazuhide Nagashima, Philip Pogge Von Strandmann, Martin Oeser, Noah Planavsky, Franck Poitrasson, John Reinfelder, Stephen Romaniello, Mathieu Roskosz, Kate Scheiderich, Kathrin Schilling, Laura Sherman, Haolan Tang, Francois Tissot, Paul Tomascak, Martin Tsz-Ki Tsui, Jim Van Orman, Xiangli Wang, Laura Wasylenki, Dominik Weiss, Stefan Weyer, Jan Wiederhold, Martin Wille, Josh Wimpenny, Wei Yang, Karen Ziegler, and many anonymous reviewers. We gratefully acknowledge the help from Don DePaolo, Valarie Espinoza-Ross, and Kryshna Avina in organizing and hosting the workshop at LBNL. We are also indebted to Ian Swainson, series editor, for all his work in producing this volume, and Alex Speer at the MSA business office, for help in the preparation of the volume and management of registrations and donations. We also thank Matt Kohn and Youxue Zhang for sharing their experience in preparing a volume and workshop. Finally, the financial support provided by Nu Instruments, Cameca and Ametek, Elemental Scientific, Geochemical Society, Savillex, Depths of the Earth, and Thermo-Fisher are highly appreciated. Fang-Zhen Teng, Seattle, Washington James Watkins, Eugene, Oregon Nicolas Dauphas, Chicago, Illinois November 2016 1529-6466/17/0082-0000$00.00

http://dx.doi.org/10.2138/rmg.2017.82.0

82

Non-Traditional Stable Isotopes Reviews in Mineralogy and Geochemistry

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FROM THE SERIES EDITOR It has been a pleasure working with the volume editors and authors on this 82nd volume of Reviews in Mineralogy and Geochemistry. Several chapters have associated supplemental figures and or tables that can be found at the MSA website. Any future errata will also be posted there. Ian P. Swainson, Series Editor Vienna, Austria

PREFACE Since the publication of Geochemistry of Non-Traditional Stable Isotopes in 2004 (volume 55 of Reviews in Mineralogy and Geochemistry), analytical techniques have significantly improved and new research directions have emerged in non-traditional stable isotope geochemistry. Our goal here is to review the current status of non-traditional isotope geochemistry from analytical, theoretical, and experimental approaches to analysis of natural samples. In particular, important applications to cosmochemistry, high-temperature geochemistry, low-temperature geochemistry, and geobiology are discussed. The aim of this volume is to provide the most comprehensive review on non-traditional isotope geochemistry for students and researchers who are interested in both the theory and applications of non-traditional stable isotope geochemistry. We take this opportunity to thank the timely contributions by authors of the individual chapters and insightful reviews from the following scientists: Bridget Bergquist, Greg Brennecka, Christopher Cloquet, Hans Eggenkamp, Toshiyuki Fujii, Sarah Gleeson, Doug Hammond, Adrianna Heimann, Richard Hervig, Fang Huang, Timm John, Tom Johnson, Abby Kavner, James Kubicki, Laura Lammers, Sheng-Ao Liu, Catherine Macris, Paul Mason, Ryan Mathur, Vasileios Mavromatis, Fred Moynier, Kazuhide Nagashima, Philip Pogge Von Strandmann, Martin Oeser, Noah Planavsky, Franck Poitrasson, John Reinfelder, Stephen Romaniello, Mathieu Roskosz, Kate Scheiderich, Kathrin Schilling, Laura Sherman, Haolan Tang, Francois Tissot, Paul Tomascak, Martin Tsz-Ki Tsui, Jim Van Orman, Xiangli Wang, Laura Wasylenki, Dominik Weiss, Stefan Weyer, Jan Wiederhold, Martin Wille, Josh Wimpenny, Wei Yang, Karen Ziegler, and many anonymous reviewers. We gratefully acknowledge the help from Don DePaolo, Valarie Espinoza-Ross, and Kryshna Avina in organizing and hosting the workshop at LBNL. We are also indebted to Ian Swainson, series editor, for all his work in producing this volume, and Alex Speer at the MSA business office, for help in the preparation of the volume and management of registrations and donations. We also thank Matt Kohn and Youxue Zhang for sharing their experience in preparing a volume and workshop. Finally, the financial support provided by Nu Instruments, Cameca and Ametek, Elemental Scientific, Geochemical Society, Savillex, Depths of the Earth, and Thermo-Fisher are highly appreciated. Fang-Zhen Teng, Seattle, Washington James Watkins, Eugene, Oregon Nicolas Dauphas, Chicago, Illinois November 2016 1529-6466/17/0082-0000$00.00

http://dx.doi.org/10.2138/rmg.2017.82.0

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TABLE OF CONTENTS

1

Non-Traditional Stable Isotopes: Retrospective and Prospective Fang-Zhen Teng, Nicolas Dauphas, James M. Watkins

INTRODUCTION ....................................................................................................................1 THE NOTATION .................................................................................................................3 GUIDELINES FOR SELECTING REFERENCE MATERIALS ............................................4 EMERGING ISOTOPE SYSTEMS .........................................................................................6 Stable potassium isotope geochemistry .........................................................................7 Titanium isotope geochemistry .....................................................................................8 Vanadium isotope geochemistry ..................................................................................10 Stable rubidium isotope geochemistry ........................................................................11 Stable strontium isotope geochemistry........................................................................11 Cadmium isotope geochemistry ..................................................................................13 Tin isotope geochemistry.............................................................................................15 Antimony isotope geochemistry ..................................................................................15 Stable tellurium isotope geochemistry ........................................................................16 Barium isotope geochemistry ......................................................................................16 Stable neodymium isotope geochemistry ....................................................................18 CONCLUSIONS.....................................................................................................................20 ACKNOWLEDGMENTS.......................................................................................................20 REFERENCES .......................................................................................................................20

2

Equilibrium Fractionation of Non-traditional Isotopes: a Molecular Modeling Perspective Marc Blanchard, Etienne Balan, Edwin A. Schauble

INTRODUCTION ..................................................................................................................27 THEORETICAL FRAMEWORK ..........................................................................................28 Equilibrium fractionation theory .................................................................................28 Approximate formula based on force constants ..........................................................33 MODELING APPROACHES .................................................................................................35 Quantum-mechanical molecular modeling..................................................................35 Theoretical studies of non-traditional stable isotope fractionation .............................37 iv

Non-Traditional Stable Isotopes Modeling isotopic properties of liquid phases.............................................................40 Beyond harmonic approximation: Path integral molecular dynamics.........................43 MÖSSBAUER AND NRIXS SPECTROSCOPY ...................................................................45 MASS-INDEPENDENT FRACTIONATION AND VARIATIONS IN MASS LAWS...................................................................................47 Variability in mass laws for common fractionations ...................................................48 Mass-independent fractionation in light elements (O and S) ......................................50 Mass-independent fractionation in non-traditional elements (Hg, Tl, and U) ............50 Mass-independent fractionation signatures in heavy elements, versus light elements ................................................................................................53 Ab initio methods for calculating field shift fractionation factors ...............................53 Isomer shifts from Mössbauer spectroscopy ...............................................................55 CONCLUSIONS.....................................................................................................................55 ACKNOWLEDGMENTS.......................................................................................................56 REFERENCES .......................................................................................................................56

3

Equilibrium Fractionation of Non-Traditional Stable Isotopes: an Experimental Perspective Anat Shahar, Stephen M. Elardo, Catherine A. Macris

INTRODUCTION ..................................................................................................................65 FACTORS INFLUENCING EQUILIBRIUM FRACTIONATION FACTORS.....................66 PROOF OF EQUILIBRIUM IN ISOTOPE EXPERIMENTS ...............................................67 Time series...................................................................................................................67 Multi-direction approach .............................................................................................68 Three-isotope exchange method ..................................................................................69 Kinetic effects ..............................................................................................................72 EXPERIMENTAL METHODS ..............................................................................................73 Low temperature experiments .....................................................................................73 High temperature, low pressure experiments ..............................................................74 High temperature and pressure experiments ...............................................................76 NRIXS and diamond anvil cell experiments ...............................................................78 POST-EXPERIMENT ANALYSIS ........................................................................................80 CONCLUSIONS.....................................................................................................................80 ACKNOWLEDGMENTS.......................................................................................................81 REFERENCES .......................................................................................................................81

4

Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions James M. Watkins, Donald J. DePaolo, E. Bruce Watson

INTRODUCTION ..................................................................................................................85 Organization of the article ...........................................................................................86 ISOTOPE FRACTIONATION BY DIFFUSION ...................................................................86 Expressions for diffusive fluxes...................................................................................87 Isotopic mass dependence of diffusion in “simple” systems.......................................87 Isotopic mass dependence of diffusion in aqueous solution .......................................88 v

Non-Traditional Stable Isotopes Isotopic mass dependence of diffusion in silicate melts .............................................90 Isotopic mass dependence of diffusion in minerals and metals...................................92 DIFFUSIVE BOUNDARY LAYERS IN THE GROWTH MEDIUM ...................................94 ISOTOPE FRACTIONATION BY COMBINED REACTION AND DIFFUSION ............102 General framework for crystal growth from an infinite solution...............................102 Crystal growth and kinetic isotope effects ................................................................105 Interpreting the model parameters .............................................................................107 Stable isotope fractionation during electroplating.....................................................110 Stable isotope fractionation of trace elements ...........................................................113 THE ROLE OF THE NEAR SURFACE OF CRYSTALS ...................................................115 The growth entrapment model (GEM) ......................................................................116 The surface reaction kinetic model (SRKM), growth entrapment model (GEM), and isotopes ............................................................................................................118 PERSPECTIVES ..................................................................................................................120 ACKNOWLEDGMENTS.....................................................................................................121 REFERENCES .....................................................................................................................121

5

In Situ Analysis of Non-Traditional Isotopes by SIMS and LA–MC–ICP–MS: Key Aspects and the Example of Mg Isotopes in Olivines and Silicate Glasses Marc Chaussidon, Zhengbin Deng, Johan Villeneuve, Julien Moureau, Bruce Watson, Frank Richter, Frédéric Moynier

INTRODUCTION ................................................................................................................127 Notations used for Mg isotopes .................................................................................128 INSTRUMENTATION FOR IN-SITU STABLE ISOTOPE ANALYSIS ............................128 MC–SIMS analysis....................................................................................................129 LA–MC–ICP–MS analysis........................................................................................129 LIMITATIONS FOR IN-SITU STABLE ISOTOPES ANALYSIS ......................................130 Limitations due to the small amount of sample analyzed .........................................131 Limitations due to matrix effects on ion yield...........................................................131 Limitations due to instrumental isotopic fractionation..............................................133 STANDARDS AND ANALYTICAL APPROACH USED FOR MG IN THE PRESENT STUDY .................................................................................................136 Set of standards studied .............................................................................................136 MC–SIMS for Mg isotopic analysis ..........................................................................137 LA–MC–ICP–MS for Mg isotopic analysis ..............................................................138 Solution MC–ICP–MS for Mg isotopic analysis ......................................................138 MAGNESIUM ION EMISSION DURING IN SITU ANALYSIS ......................................140 Fundamental differences for Mg ion yield between SIMS and laser ablation ICP–MS............................................................................................140 Possible origin of the complex matrix effects on ion yield for SIMS ......................143 MAGNESIUM INSTRUMENTAL ISOTOPIC FRACTIONATION ...................................145 Similarities and differences for Mg instrumental isotopic fractionation between SIMS and laser ablation ICP–MS ............................................................145 Matrix effects during ionization of solutions in MC–ICP–MS ................................147 Matrix effects specific to in situ analysis ..................................................................149 MEASUREMENT OF THE THREE MAGNESIUM ISOTOPES.......................................152 vi

Non-Traditional Stable Isotopes The need for high-precision in situ three Mg isotopes analysis in cosmochemistry .152 The question of potential isobaric interferences........................................................153 The question of the mass fractionation law used to correct for instrumental isotopic fractionation ..............................................................................................154 Mg instrumental mass fractionation law for MC–SIMS analyses.............................155 Mg instrumental mass fractionation law for LA–MC–ICP–MS analyses.................158 SUMMARY AND PERSPECTIVES ...................................................................................158 ACKNOWLEDGMENTS.....................................................................................................159 REFERENCES .....................................................................................................................159

6

Lithium Isotope Geochemistry Sarah Penniston-Dorland,, Xiao-Ming Liu, Roberta L. Rudnick

INTRODUCTION ................................................................................................................165 LITHIUM SYSTEMATICS ..................................................................................................167 Li in minerals.............................................................................................................168 Li partitioning ............................................................................................................168 Equilibrium Isotope Fractionation.............................................................................170 Diffusion and kinetic isotopic fractionation ..............................................................174 METHODS ...........................................................................................................................176 Whole rock analyses ..................................................................................................176 In situ analyses ..........................................................................................................177 EXTRATERRESTRIAL LITHIUM RESERVOIRS ............................................................178 The interstellar medium and the Sun .........................................................................178 Meteorites and their components...............................................................................179 Moon .........................................................................................................................181 TERRESTRIAL LITHIUM RESERVOIRS .........................................................................181 Mantle peridotites ......................................................................................................181 Basalts .......................................................................................................................188 Arc lavas ....................................................................................................................190 Continental crust........................................................................................................191 Seawater ....................................................................................................................192 Rivers .........................................................................................................................193 Lakes..........................................................................................................................193 Groundwater ..............................................................................................................194 Hydrothermal fluids ...................................................................................................194 IGNEOUS PROCESSES ......................................................................................................195 Differentiation ...........................................................................................................195 Eruptive processes .....................................................................................................196 METAMORPHIC PROCESSES...........................................................................................197 Dehydration ...............................................................................................................197 Redistribution of Li through fluid infiltration ............................................................198 Diffusion ....................................................................................................................199 CONTINENTAL WEATHERING PROCESSES .................................................................199 Weathering profiles ....................................................................................................201 Rivers .........................................................................................................................201 LITHIUM AS A TRACER OF CONTINENTAL WEATHERING THROUGH TIME..............................................................................................................204 vii

Non-Traditional Stable Isotopes FUTURE DIRECTIONS ......................................................................................................205 Weathering processes and Li fractionation experiments ...........................................205 Continental weathering through time ........................................................................205 Geospeedometry ........................................................................................................206 ACKNOWLEDGMENTS.....................................................................................................206 REFERENCES .....................................................................................................................206

7

Magnesium Isotope Geochemistry Fang-Zhen Teng

INTRODUCTION ................................................................................................................219 MAGNESIUM ISOTOPIC ANALYSIS ...............................................................................221 Nomenclature ............................................................................................................221 Standard and reference materials...............................................................................222 Instrumental Analysis ................................................................................................229 Sample preparation ....................................................................................................229 MAGNESIUM ISOTOPIC SYSTEMATICS OF EXTRATERRESTRIAL RESERVOIRS .....................................................................230 Magnesium isotopic composition of chondrites ........................................................231 Magnesium isotopic composition of differentiated meteorites .................................232 Magnesium isotopic composition of the Moon .........................................................232 MAGNESIUM ISOTOPIC SYSTEMATICS OF THE MANTLE .......................................234 Mantle xenoliths ........................................................................................................234 Oceanic basalts ..........................................................................................................236 Abyssal peridotites and ophiolites .............................................................................237 Continental basalts ....................................................................................................239 MAGNESIUM ISOTOPIC SYSTEMATICS OF THE OCEANIC CRUST, CONTINENTAL CRUST AND HYDROSPHERE...........................................................241 Magnesium isotopic composition of the oceanic crust..............................................241 Magnesium isotopic composition of the continental crust ........................................243 Magnesium isotopic composition of the hydrosphere ...............................................247 MAGNESIUM ISOTOPIC SYSTEMATICS OF CARBONATES ......................................250 Abiogenic carbonates ................................................................................................250 Biogenic carbonates...................................................................................................254 Carbonate precipitation experiments and theoretical calculations ............................255 BEHAVIOR OF MAGNESIUM ISOTOPES DURING MAJOR GEOLOGICAL PROCESSES ............................................................256 Behavior of Mg isotopes during biological processes ...............................................256 Behavior of Mg isotopes during continental weathering ..........................................257 Behavior of Mg isotopes during magmatic differentiation .......................................263 Behaviors of Mg isotopes during metamorphic dehydration ....................................264 HIGH-TEMPERATURE MAGNESIUM ISOTOPE FRACTIONATION ...........................267 High-temperature equilibrium inter-mineral Mg isotope fractionation.....................267 Diffusion-driven kinetic Mg isotope fractionation ....................................................270 APPLICATIONS AND FUTURE DIRECTIONS ................................................................276 ACKNOWLEDGMENTS.....................................................................................................278 REFERENCES .....................................................................................................................278 viii

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Silicon Isotope Geochemistry Franck Poitrasson

ELEMENT PROPERTIES ....................................................................................................289 NOMENCLATURE, REFERENCE MATERIALS AND ANALYTICAL TECHNIQUES.........................................................................................290 ELEMENTAL AND ISOTOPIC ABUNDANCES IN MAJOR RESERVOIRS ..................294 Extraterrestrial reservoirs ..........................................................................................294 Terrestrial reservoirs ..................................................................................................301 ELEMENTAL AND ISOTOPIC BEHAVIORS DURING MAJOR GEOLOGICAL PROCESSES ............................................................317 Diffusion, condensation and evaporation ..................................................................317 Igneous processes ......................................................................................................322 Metamorphic processes .............................................................................................325 Low temperature processes .......................................................................................325 Biological processes ..................................................................................................330 IMPORTANT IMPLICATIONS AND FUTURE DIRECTIONS ........................................333 ACKNOWLEDGMENTS.....................................................................................................336 REFERENCES .....................................................................................................................337

9

Chlorine Isotope Geochemistry Jaime D. Barnes, Zachary D. Sharp

INTRODUCTION ................................................................................................................345 CHLORINE ISOTOPE NOMENCLATURE AND STANDARDS......................................346 CHLORINE ISOTOPE ANALYTICAL METHODS ..........................................................346 Isotope ratio mass spectrometry (IRMS)...................................................................346 Thermal ionization mass spectrometry (TIMS) ........................................................347 Secondary ion mass spectrometry (SIMS) ................................................................348 Laser ablation inductively coupled plasma mass spectrometry (LA–ICP–MS)........348 ISOTOPIC FRACTIONATION ............................................................................................348 Equilibrium Cl isotope fractionation—theoretical constraints ..................................348 Equilibrium Cl isotope fractionation—experimental constraints ..............................349 Kinetic Cl isotope fractionation—Cl loss ................................................................351 CHLORINE ISOTOPIC COMPOSITION OF VARIOUS GEOLOGIC RESERVOIRS ..............................................................................351 Mantle/OIB/mantle derived material .........................................................................351 Seawater and seawater-derived chloride....................................................................353 Sediments ..................................................................................................................355 Altered Oceanic Crust (AOC) ...................................................................................355 Serpentinites ..............................................................................................................357 Perchlorates ...............................................................................................................357 Extraterrestrial Materials ...........................................................................................358 CHLORINE ISOTOPES AS A TRACER ............................................................................364 Tracer through subduction zones...............................................................................364 Crustal fluids .............................................................................................................365 Tracer in ore deposits ................................................................................................368 ix

Non-Traditional Stable Isotopes Environmental ...........................................................................................................370 SUMMARY ..........................................................................................................................370 ACKNOWLEDGMENTS.....................................................................................................371 REFERENCES .....................................................................................................................371

10

Chromium Isotope Geochemistry Liping Qin, Xiangli Wang

INTRODUCTION ................................................................................................................379 Chemical properties of Cr .........................................................................................379 Research History of the Cr isotopic System ..............................................................380 ANALYTICAL METHODS AND NOTATION ...................................................................382 Analytical methods ....................................................................................................382 Notation .....................................................................................................................385 CHROMIUM ISOTOPE COSMOCHEMISTRY .................................................................386 53 Mn–53Cr short-lived chronometer ...........................................................................386 54 Cr anomalies............................................................................................................387 CHROMIUM ISOTOPIC FRACTIONATION IN HIGH-TEMPERATURE SETTINGS ..........................................................................388 Bulk silicate earth and meteorites..............................................................................388 Serpentinization and metamorphism .........................................................................389 MECHANISMS OF CR ISOTOPIC FRACTIONATION IN LOW-TEMPERATURE SETTINGS ............................................................................390 Reduction...................................................................................................................390 Equilibrium Cr isotopic fractionation and Cr(III)–Cr(VI) isotope exchange in aqueous systems .................................................................................................397 Oxidation ...................................................................................................................398 Adsorption .................................................................................................................398 Coprecipitation ..........................................................................................................398 CHROMIUM ISOTOPIC VARIATIONS IN SURFACIAL ENVIRONMENTS .................399 Groundwater ..............................................................................................................399 Weathering systems ...................................................................................................399 Rivers and seawater ...................................................................................................400 Cr isotope mass balance ............................................................................................401 The Cr isotope system as a paleo-redox proxy..........................................................403 CONCLUDING REMARKS AND OUTLOOK ..................................................................407 ACKNOWLEDGEMENT ....................................................................................................408 REFERENCES .....................................................................................................................408

11

Iron Isotope Systematics Nicolas Dauphas, Seth G. John, Olivier Rouxel

INTRODUCTION ................................................................................................................415 METHODOLOGY ...............................................................................................................417 Rocks and solid samples............................................................................................418 Water samples ............................................................................................................423 In situ analyses ..........................................................................................................424 Isotopic anomalies and mass-fractionation laws .......................................................428 x

Non-Traditional Stable Isotopes KINETIC AND EQUILIBRIUM FRACTIONATION FACTORS.......................................429 Kinetic processes .......................................................................................................430 Equilibrium processes ...............................................................................................433 IRON ISOTOPES IN COSMOCHEMISTRY ......................................................................451 Nucleosynthetic anomalies and iron-60 ....................................................................451 Overview of iron isotopic compositions in extraterrestrial material .........................454 HIGH-TEMPERATURE GEOCHEMISTRY ......................................................................456 Partial mantle melting................................................................................................462 Impact evaporation and core formation .....................................................................464 Fractional crystallization, fluid exsolution, immiscibility, and thermal (Soret) diffusion ..................................................................................................................466 A new tool to improve on geospeedometry reconstructions in igneous petrology....469 IRON BIOGEOCHEMISTRY ..............................................................................................470 Microbial cycling of Fe isotopes ...............................................................................470 Fe isotopes in plants, animals, and humans...............................................................472 FLUID–ROCK INTERACTIONS ........................................................................................473 High- and low-temperature alteration processes at the seafloor ................................473 Rivers and soils..........................................................................................................475 Mineral deposits ........................................................................................................475 IRON BIOGEOCHEMICAL CYCLING IN THE MODERN OCEAN ..............................477 The importance of iron in the global ocean ...............................................................477 Sources and sinks for Fe in the ocean .......................................................................478 Using Fe isotopes to trace sources of Fe in the oceans .............................................482 Internal cycling of Fe isotopes within the ocean .......................................................482 THE GEOLOGICAL RECORD AND PALEOCEANOGRAPHIC APPLICATIONS ........483 The ferromanganese crust record ..............................................................................483 Oceanic anoxic events ...............................................................................................485 The Precambrian record ............................................................................................486 The archive of iron formations ..................................................................................486 Black Shales and Sedimentary Pyrite Archives .........................................................490 CONCLUSION .....................................................................................................................492 ACKNOWLEDGMENTS.....................................................................................................492 REFERENCES .....................................................................................................................493

12

The Isotope Geochemistry of Ni Tim Elliott, Robert C. J. Steele

INTRODUCTION ................................................................................................................511 Notation .....................................................................................................................511 NUCLEOSYNTHETIC NI ISOTOPIC VARIATIONS ........................................................513 EXTINCT 60Fe AND RADIOGENIC 60Ni ...........................................................................520 MASS-DEPENDENT Ni ISOTOPIC VARIABILITY .........................................................526 Magmatic systems .....................................................................................................526 Weathering and the hydrological cycle .....................................................................532 Biological systems .....................................................................................................534 OUTLOOK ...........................................................................................................................536 ACKNOWLEDGMENTS.....................................................................................................537 REFERENCES .....................................................................................................................537 xi

Non-Traditional Stable Isotopes

13

The Isotope Geochemistry of Zinc and Copper Frédéric Moynier, Derek Vance, Toshiyuki Fujii, Paul Savage

INTRODUCTION ................................................................................................................543 METHODS ...........................................................................................................................545 ZINC AND COPPER ISOTOPE FRACTIONATION FACTORS FROM AB INITIO METHODS ........................................................................................548 ZINC AND COPPER IN EXTRA-TERRESTRIAL SAMPLES AND IGNEOUS ROCKS ..................................................................................................557 ZINC AND COPPER IN LOW TEMPERATURE GEOCHEMISTRY ...............................566 Experimental constraints on fractionation mechanisms ............................................568 Cu–Zn isotopes in the weathering–soil–plant system ...............................................576 The oceans: inputs, outputs and internal cycling of Cu and Zn isotopes ..................581 Cu and Zn isotopes in the Anthropocene...................................................................589 ACKNOWLEDGMENTS.....................................................................................................591 REFERENCES .....................................................................................................................591

14

Germanium Isotope Geochemistry Olivier J. Rouxel, Béatrice Luais

INTRODUCTION ................................................................................................................601 METHODS ...........................................................................................................................602 Early methods to measure Ge isotope ratios .............................................................602 State of the art analytical methods.............................................................................603 Sample dissolution issues ..........................................................................................605 Chemical purification of samples ..............................................................................606 Hydride generation (HG) MC-ICPMS .....................................................................606 Interference issues .....................................................................................................607 Notation .....................................................................................................................607 Analytical precision ..................................................................................................608 Ge isotope standards and reference materials ..........................................................609 THEORETICAL CONSIDERATIONS AND EXPERIMENTAL CALIBRATIONS .........612 Equilibrium fractionation factors ..............................................................................612 Kinetic processes .......................................................................................................613 Diffusion of Ge in silicate melts ................................................................................614 HIGH-TEMPERATURE GEOCHEMISTRY ......................................................................616 Fundamentals of Ge high-temperature geochemistry................................................616 Cosmochemistry of Ge isotopes ................................................................................621 GERMANIUM ISOTOPE SYSTEMATICS IN IGNEOUS, MANTLE-DERIVED ROCKS, AND METAMORPHIC ROCKS ...................................626 The Ge isotopic composition of the Earth silicate reservoirs ....................................626 Germanium recycling into the mantle: an attempt to evaluate mantle homogeneity 628 Ore deposits ...............................................................................................................629 LOW-TEMPERATURE GEOCHEMISTRY ........................................................................631 Fundamentals of Ge low-temperature geochemistry .................................................631 Germanium isotope systematics in low-temperature marine environments ..............634 Ge isotope fractionation during low temperature weathering ...................................639 xii

Non-Traditional Stable Isotopes Germanium isotope systematics of hydrothermal waters ..........................................639 A preliminary oceanic Ge budget ..............................................................................641 The potential for paleoceanography and the rock record ..........................................644 CONCLUSION .....................................................................................................................647 ACKNOWLEDGMENTS.....................................................................................................647 REFERENCES .....................................................................................................................648

15

Selenium Isotopes as a Biogeochemical Proxy in Deep Time Eva E. Stüeken

INTRODUCTION AND OVERVIEW .................................................................................657 NOMENCLATURE, REFERENCE MATERIALS AND ANALYTICAL TECHNIQUES 659 ELEMENTAL AND ISOTOPIC ABUNDANCES IN MAJOR RESERVOIRS ..................661 Terrestrial and extraterrestrial igneous terrestrial reservoirs .....................................661 Reservoirs at the Earth’s surface ...............................................................................663 SELENIUM IN BIOLOGY ..................................................................................................666 ISOTOPIC FRACTIONATION PATHWAYS.......................................................................667 GEOBIOLOGICAL APPLICATIONS .................................................................................670 Developing a mass balance for the modern ocean ....................................................670 Implications and predictions......................................................................................673 Selenium isotopes in deep time .................................................................................674 CONCLUSIONS AND FUTURE DIRECTIONS ................................................................677 ACKNOWLEDGMENTS.....................................................................................................677 REFERENCES .....................................................................................................................677

16

Good Golly, Why Moly? The Stable Isotope Geochemistry of Molybdenum Brian Kendall, Tais W. Dahl, Ariel D. Anbar

INTRODUCTION ................................................................................................................683 ANALYTICAL CONSIDERATIONS ..................................................................................684 Data reporting ............................................................................................................684 Chemical separation ..................................................................................................685 Mass Spectrometry ....................................................................................................686 CHEMICAL AND BIOLOGICAL CONTEXT ...................................................................688 Aqueous geochemistry ..............................................................................................688 Biology ......................................................................................................................690 FRACTIONATION FACTORS ...........................................................................................692 Adsorption to Mn oxides ...........................................................................................692 Adsorption to Fe oxides and oxyhydroxides .............................................................693 Sulfidic species ..........................................................................................................694 Biological processes ..................................................................................................694 High-temperature melt systems .................................................................................695 MOLYBDENUM ISOTOPES IN MAJOR RESERVOIRS..................................................696 Meteorites ..................................................................................................................696 The mantle and crust .................................................................................................697 The oceans .................................................................................................................699 Lakes..........................................................................................................................706 xiii

Non-Traditional Stable Isotopes APPLICATION TO OCEAN PALEOREDOX .....................................................................706 Local depositional conditions ....................................................................................707 Reconstructing the oceanic Mo isotope mass balance ..............................................708 Inferring seawater 98Mo from sedimentary archives................................................710 Tracing atmosphere–ocean oxygenation using Mo isotopes .....................................712 Part 1: Searching for free O2 in the Archean surface environment ............................712 Part 2: Tracing global ocean oxygenation in the post-GOE world ............................714 APPLICATION TO NATURAL RESOURCES ...................................................................717 Ore deposits ...............................................................................................................717 Petroleum systems .....................................................................................................722 Anthropogenic tracing ...............................................................................................722 CONCLUSIONS...................................................................................................................723 ACKNOWLEDGMENTS.....................................................................................................724 REFERENCES .....................................................................................................................724

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Recent Developments in Mercury Stable Isotope Analysis Joel D. Blum, Marcus W. Johnson

INTRODUCTION ................................................................................................................733 Element properties .....................................................................................................734 Hg isotope nomenclature ...........................................................................................734 Reference materials ...................................................................................................735 Analytical advances ...................................................................................................735 Isobaric and molecular interferences .........................................................................737 Reagent blanks and sample carryover .......................................................................740 Matrix effects and the need to remove matrix materials ...........................................741 Recommendations for analyses .................................................................................744 ISOTOPIC ABUNDANCES IN MAJOR RESERVOIRS ....................................................745 Isotopic behaviors during major geological processes ..............................................748 NEW FRONTIERS IN HG ISOTOPE ANALYSIS..............................................................749 Isotopic composition of methyl mercury...................................................................749 Separation of Hg from low concentration waters ......................................................750 Separation of Hg from atmospheric gases .................................................................750 Even-mass MIF .........................................................................................................751 FUTURE DIRECTIONS ......................................................................................................752 ACKNOWLEDGMENTS.....................................................................................................753 REFERENCES .....................................................................................................................753

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Investigation and Application of Thallium Isotope Fractionation Sune G. Nielsen, Mark Rehkämper, Julie Prytulak

ABSTRACT ..........................................................................................................................759 INTRODUCTION ................................................................................................................760 METHODOLOGY ...............................................................................................................762 Mass spectrometry .....................................................................................................762 Chemical separation of thallium................................................................................762 Measurement uncertainties and standards .................................................................764 xiv

Non-Traditional Stable Isotopes THALLIUM ISOTOPE VARIATION IN EXTRATERRESTRIAL MATERIALS .............765 The 205Pb–205Tl decay system.....................................................................................765 Chondritic meteorites ................................................................................................766 Iron meteorites ...........................................................................................................767 Limitations of the 205Pb–205Tl chronometer ...............................................................770 THALLIUM ISOTOPE COMPOSITION OF THE SOLID EARTH ..................................771 The primitive mantle .................................................................................................771 The continental crust .................................................................................................771 THALLIUM ISOTOPE COMPOSITION OF SURFACE RESERVOIRS ..........................772 Volcanic degassing ....................................................................................................772 Weathering and riverine transport of Tl.....................................................................773 Anthropogenic mobilization of Tl .............................................................................774 The isotope composition of seawater ........................................................................775 THE MARINE MASS BALANCE OF THALLIUM ISOTOPES .......................................776 Thallium isotopes in marine input fluxes ..................................................................776 Thallium isotope compositions of marine output fluxes ...........................................780 CAUSES OF THALLIUM ISOTOPE FRACTIONATION..................................................782 APPLICATIONS OF THALLIUM ISOTOPES ...................................................................784 Studies of Tl isotopes in Fe–Mn crusts .....................................................................784 Calculation of hydrothermal fluid fluxes using Tl isotopes in the ocean crust..........786 High temperature terrestrial applications ..................................................................787 FUTURE DIRECTIONS AND OUTLOOK ........................................................................791 REFERENCES .....................................................................................................................793

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Uranium Isotope Fractionation Morten B. Andersen, Claudine H. Stirling, Stefan Weyer

INTRODUCTION ................................................................................................................799 Uranium occurrence and properties...........................................................................799 Uranium isotopes .......................................................................................................800 URANIUM ISOTOPE DETERMINATIONS ......................................................................803 Historical overview of 238U/235U measurements.........................................................803 Chemical preparation of U and mass spectrometric corrections ...............................804 Anthropogenic U contamination ...............................................................................804 238 U/235U nomenclature ..............................................................................................805 EXPERIMENTAL EVIDENCE FOR URANIUM ISOTOPE FRACTIONATION PROCESSES ......................................................................................................................806 Experimental studies for nuclear 235U enrichment.....................................................806 Uranium isotopes and the nuclear field shift .............................................................806 Experimental evidence for kinetic and equilibrium 238U/235U fractionation ..............807 238 U/235U IN COSMOCHEMISTRY .....................................................................................809 Uranium isotopic anomalies and the search for extant 247Cm ...................................809 Uranium isotope fractionation unrelated to 247Cm decay ..........................................813 Pb–Pb chronometer ...................................................................................................815 URANIUM ISOTOPE SYSTEMATICS IN HIGH TEMPERATURE ENVIRONMENTS ON EARTH .......................................................................................817 Bulk Earth 238U/235U ...................................................................................................817 The mantle .................................................................................................................818 xv

Non-Traditional Stable Isotopes The continental crust .................................................................................................819 URANIUM ISOTOPES IN ORE DEPOSITS ......................................................................820 Uranium ore types .....................................................................................................821 Uranium isotope fractionation in ore deposits...........................................................821 Uranium isotopes as tracers of U mine remediation .................................................824 NEAR-SURFACE U CYCLING AND THE MARINE 238U/235U MASS BALANCE .........825 The 238U/235U in rivers and groundwater ....................................................................826 The 238U/235U of seawater ...........................................................................................827 The 238U/235U in reducing sediments. .........................................................................827 The 238U/235U in marine carbonates............................................................................831 The 238U/235U of altered oceanic crust ........................................................................832 The 238U/235U of ferromangenese oxides....................................................................833 Isotopic constraints on the marine U cycle................................................................834 U ISOTOPES AS A PALEO-REDOX PROXY ....................................................................837 Early Earth redox evolution .......................................................................................838 Neo-Proterozoic and Phanerozoic .............................................................................839 OUTLOOK ...........................................................................................................................841 ACKNOWLEDGMENTS.....................................................................................................843 REFERENCES .....................................................................................................................843

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Medical Applications of Isotope Metallomics Francis Albarède, Philippe Télouk, Vincent Balter

INTRODUCTION ................................................................................................................851 THE ISOTOPE EFFECT ......................................................................................................853 AN OVERVIEW OF Ca, Fe, Zn, Cu, AND S BIOCHEMISTRY AND HOMEOSTASIS ..859 Calcium .....................................................................................................................859 Iron ............................................................................................................................860 Zinc ............................................................................................................................861 Copper .......................................................................................................................862 Sulfur .........................................................................................................................863 ISOTOPE COMPOSITIONS OF Fe–Zn–Cu–S IN THE BLOOD OF HEALTHY INDIVIDUALS ........................................................................................864 CALCIUM AND BONE LOSS ............................................................................................867 GENETIC AND INFECTIOUS DISEASES ........................................................................868 ISOTOPE METALLOMICS IN CANCER ..........................................................................869 ASSESSING THE POTENTIAL OF METAL ISOTOPES AS BIOMARKERS.................875 COMPARTMENTALIZED MODELS OF CELLULAR HOMEOSTASIS ........................877 PERSPECTIVES ..................................................................................................................878 ACKNOWLEDGMENTS.....................................................................................................879 REFERENCES .....................................................................................................................879

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Reviews in Mineralogy & Geochemistry Vol. 82 pp. 1-26, 2017 Copyright © Mineralogical Society of America

Non-Traditional Stable Isotopes: Retrospective and Prospective Fang-Zhen Teng Isotope Laboratory Department of Earth and Space Sciences University of Washington Seattle WA 98195 USA [email protected]

Nicolas Dauphas Origins Lab Department of the Geophysical Sciences and Enrico Fermi Institute The University of Chicago Chicago IL 60637 USA [email protected]

James M. Watkins Department of Earth Sciences University of Oregon Eugene OR 97403 USA [email protected]

INTRODUCTION Traditional stable isotope geochemistry involves isotopes of light elements such as H, C, N, O, and S, which are measured predominantly by gas-source mass spectrometry (Valley et al. 1986; Valley and Cole 2001). Even though Li isotope geochemistry was developed in 1980s based on thermal ionization mass spectrometry (TIMS) (Chan 1987), the real flourish of so-called nontraditional stable isotope geochemistry was made possible by the development of multi-collector inductively coupled plasma mass spectrometry (MC-ICPMS) (Halliday et al. 1995; Marechal et al. 1999). Since then, isotopes of both light (e.g., Li, Mg) and heavy (e.g., Tl, U) elements have been routinely measured at a precision that is high enough to resolve natural variations (Fig. 1). The publication of RIMG volume 55 (Geochemistry of Non-Traditional Stable Isotopes) in 2004 was the first extensive review of Non-Traditional Stable Isotopes summarizing the advances in the field up to 2003 (Johnson et al. 2004). When compared to traditional stable isotopes, the non-traditional stable isotopes have several distinctive geochemical features: 1) as many of these elements are trace elements, their concentrations vary widely in different geological reservoirs; 2) these elements range from highly volatile (e.g., Zn and K) to refractory (e.g., Ca and Ti); 3) many of these elements are redox-sensitive; 4) many of them are biologically active; 5) the bonding environments, especially for the metal elements, are different from those of H, C, N, O and S; and finally, 6) many of these elements have high atomic numbers and more than two stable isotopes. These features make the different elements susceptible to different fractionation mechanisms, and by extension, make them unique tracers of different cosmochemical, geological and biological processes, as highlighted throughout this volume. 1529-6466/17/0082-0001$05.00

http://dx.doi.org/10.2138/rmg.2017.82.1

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Teng, Dauphas, & Watkins

Figure 1. Non-traditional stable isotope systems covered in this volume.

Large variations have been documented in both natural samples and laboratory experiments for non-traditional stable isotopes (Fig. 2). These studies suggest that the following factors control the degree of isotope fractionation in non-traditional stable isotopes during various processes: the relative mass difference between isotopes, change of the oxidation state, biological sensitivity, and volatility. Among these elements, Li displays the largest isotopic variation in terrestrial samples, and since Li is not volatile during geological processes and is not sensitive to redox reactions and biological processes, the large isotope fractionation is controlled mainly by the large relative mass difference (Penniston-Dorland et al. 2017). For many of the other elements, other factors may be equally, if not more, important. For example, Cl exhibits the

Figure 2. The terrestrial isotopic variation vs. the relative mass difference for non-traditional stable isotopes covered in this volume (Anderson et al. 2017; Barnes and Sharp 2017; Blum and Johnson 2017; Dauphas et al. 2017; Elliott and Steele 2017; Kendall et al. 2017; Moynier et al. 2017; Nielsen et al. 2017; Penniston-Dorland et al. 2017; Poitrasson 2017; Qin and Wang 2017; Rouxel and Luais 2017; Stueken 2017; Teng 2017). The Dmi−j = mi − mj, where i and j represent the two isotopes that are used to report the isotopic variation, with i > j.

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second largest isotopic variation, which is due to kinetic isotope fractionation during volcanic degassing (Barnes and Sharp 2017). Selenium isotopes also show large isotopic variations, but this reflects redox-controlled Se isotope fractionation (Stueken 2017), while the large Hg isotopic variations are mainly associated with biological processes (Blum and Johnson 2017). In this chapter, we discuss guidelines and recommendations for reporting non-traditional stable isotopic data and choosing reference materials. We then provide brief introductions to some of the emerging isotope systems that are not covered as individual chapters in this volume. As Ca isotope geochemistry has been recently reviewed in the book series on Advances in Isotope Geochemistry (Gussone et al. 2016), and a similar book is also in preparation for B isotope geochemistry, both Ca and B isotopes are not discussed in this volume. For the basics of stable isotope geochemistry, we recommend prior RIMG volumes (Valley et al. 1986; Valley and Cole 2001; Johnson et al. 2004; MacPherson 2008) and stable isotope geochemistry textbooks (Criss 1999; Faure and Mensing 2005; Sharp 2007; Hoefs 2009).

THE δ NOTATION The isotopic composition of a sample is commonly reported relative to an international standard as defined by the δ notation. For example, for Mg isotopes:

 (26 Mg/ 24 Mg)sample  = δ26 Mg  26 − 1 × 1000 24  ( Mg/ Mg)DSM3 

(1)

where DSM3 is the international standard for reporting Mg isotopic data (Galy et al. 2003). This definition of δ value has been used extensively for both non-traditional stable isotopes (e.g., this volume, Johnson et al. 2004) and traditional stable isotopes (e.g., Valley et al. 1986). Recently, a new δ notation without the factor of 1000 has been recommended by the IUPAC for expression of stable isotope ratios (Coplen 2011). For example, the new guideline would suggest a definition of δ for Mg isotopes as:  (26 Mg/ 24 Mg)sample  = δ26 Mg  26 − 1 24   ( Mg/ Mg)DSM3

(2)

The rationale is that when isotopic data are reported, the ‰ symbol is commonly placed following the value. Then, in the case that (26Mg/24Mg)sample/(26Mg/24Mg)DSM3 = 1.00025, the d26Mg should be equal to + 0.25 in the traditional δ notation i.e., d26Mg = +0.25. If the ‰ symbol is added after the value i.e., d26Mg = +0.25‰, then mathematically, this means d26Mg = +0.25‰ = +0.00025. In other words, (26Mg/24Mg)sample/(26Mg/24Mg)DSM3 = 1.00000025. If the IUPAC δ notation is adopted, then when the (26Mg/24Mg)sample/(26Mg/24Mg)DSM3 = 1.00025, the d26Mg = +0.00025 = +0.25‰. While mathematically rigorous, the IUPAC recommendation goes against practices in the field. Retaining the factor of 1000 in the δ notation is, in the view of many in the community, critical to distinguish the δ, ε, and µ notations, which can be used concurrently in a paper. In the literature on traditional and non-traditional stable isotopes, the δ notation (1) with the factor of 1000 still prevails. For the purpose of being mathematically correct, an alternative to the IUPAC recommendation is to keep the traditional δ notation with the factor of 1000 and omit the ‰ symbol after the δ value, e.g., d26Mg = +0.25, which means (26Mg/24Mg)sample/ (26Mg/24Mg)DSM3 = 1.00025. This expression is also consistent with the usage of the ε notation where the part per 10000 is not added after any ε value.

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Another alternative is to adopt the original δ notation as defined by Craig (1957). For Mg isotopes:  (26 Mg/ 24 Mg)sample  = δ26 Mg(‰)  26 − 1 × 1000 24  ( Mg/ Mg)DSM3 

(3)

which would read “d26Mg expressed in ‰ is equal to [(26Mg/24Mg)sample/ (26Mg/24Mg)DSM3 − 1] × 1000”. Under this definition, it is also mathematically correct to place the ‰ symbol after the value, e.g., d26Mg = +0.25‰, which means (26Mg/24Mg)sample/ (26Mg/24Mg)DSM3 = 1.00025. Although both notations (2) and (3) are mathematically correct, we recommend the notation (3) as it ensures that new publications are consistent with past practices. More important, it is the original definition of δ notation since the beginning of stable isotope geochemistry (Craig 1957). We did not enforce this notation in the present volume but recommend it be used in future publications. Regardless of which notation is used, it is still correct, as often done by the community, to write sentences like “the 26Mg/24Mg varies 1‰ in water samples from Chicago and Seattle” or “The water sample from Chicago is enriched in heavy Mg isotopes by 1‰ relative to that in Seattle”, or “The water sample from Chicago is 1‰ heavier than that from Seattle”.

GUIDELINES FOR SELECTING REFERENCE MATERIALS The field of non-traditional stable isotope geochemistry is confronted with the same issues as traditional stable isotope geochemistry regarding the selection of isotope reference materials (IRMs; Carignan et al. 2004; Vogl and Pritzkow 2010). The recurring problem is that standards used in laboratories either run out or are not readily available. This is the case for reference materials distributed by official organizations as well as in-house standards. For instance, IRMM-014, which is used in iron isotope geochemistry and distributed by the Institute for Reference Materials and Measurements (IRMM), is no longer available. Similarly, JMC Lyon Zn and DSM3, which are used in Zn and Mg isotope geochemistry, are not readily available. These issues need to be addressed because systematic errors arise when measurements are carried out against secondary standard solutions. Below, we propose some guidelines for selecting isotopic reference materials used in isotope geochemistry. These are informed by discussions with members of the community as well as our analysis of best practices in traditional and non-traditional stable isotope geochemistry. Guideline #1: The isotopic composition of the reference material should be demonstrably homogeneous given current analytical precision and its preparation should be such that future isotopic analyses will be unlikely to reveal isotopic heterogeneities when the precision improves. We, as a community, should be forward-thinking in designing preparation protocols that minimize the possibility that reference materials will prove to be heterogeneous as analytical capabilities improve. The purified Mg metal isotopic standard NIST SRM 980, which was made and distributed by National Institute of Standards and Technology (NIST), was deemed to be isotopically homogeneous when it was created in 1966 (Catanzaro et al. 1966). However, subsequent higher precision measurements showed that it was isotopically heterogeneous, and it was replaced by a Mg solution made from dissolution of pure Mg metal from Dead Sea Magnesium Ltd, i.e., DSM3 (Galy et al. 2003). Some steps can be taken to ensure that the maximum level of homogeneity is achieved (i.e., stable solutions, quenched glasses, stable metal sheets or bars are likely to be isotopically homogeneous to a high degree). The processes that can

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potentially cause heterogeneities in isotopic composition and should be avoided are evaporation, chemical diffusion, the Soret effect, and precipitation/crystallization (e.g., Richter et al. 2009). Guideline #2: The reference material should be pure elements or chemical compounds that are either in diluted acids or can be easily dissolved into diluted acids. Any unnecessary processing performed in the lab has the potential to induce systematic errors. For example, incomplete digestion and precipitation can induce isotope fractionation, as can chemical purification of analytes in the lab. Although some of these issues can be mitigated using a double-spike approach, it is advantageous to have the reference material in a pure form or a form in which the other elements can be quantitatively removed (e.g., by drying after digestion). Guideline #3: The reference material should have an isotopic composition that falls within the range of natural variability, and ideally is representative or similar to a major geological reservoir. For non-traditional stable isotope systems, Guideline #2 imposes the reference material be purified by a third party, often at an industrial scale. A benefit is that it ensures that the reference material is available in large quantities and is unlikely to be exhausted. The process of purification can, however, induce significant isotope fractionation, such that the synthetic material can have extreme isotopic composition and may not be representative of any geochemically relevant reservoir. This can lead to all natural isotopic compositions being systematically shifted either to the negative or positive side. Therefore, the reference material should have an isotopic composition that falls within the range of natural variability and ideally is representative of a major geological reservoir for the element investigated. This can be achieved by first surveying aliquots of industrially purified material to find a batch whose isotopic composition approaches that of a geologically relevant reservoir. For example, IRMM-014 coincidentally has an iron isotopic composition that is indistinguishable from chondrites (Craddock and Dauphas 2011). Guideline #4: The isotopic composition of the reference material should be characterized at high-precision for all its isotopes to ensure that no measurable anomalies are present that would complicate studies of mass-independent effects and mass-fractionation laws. To first order, isotopic variations follow the rules of mass-dependent fractionation, meaning that the variations in isotopic ratios scale as the differences in mass of the isotopes involved. However, it is now possible to discern clear departures from mass-dependent fractionation produced by nuclear field-shift or magnetic effects, and it is also possible to precisely define the laws of mass dependent-fractionation (Dauphas and Schauble 2016). Large-scale purification processes that would fractionate isotopes would impart a certain mass-fractionation law that would complicate comparison between naturally occurring mass fractionation laws. Furthermore, it has been shown previously that some purification processes, notably the Mond-process used in purifying Ni, can create spurious isotopic anomalies (Steele et al. 2011). Characterization of mass-independent effects and mass-fractionation laws is a growing field in geochemistry (Dauphas and Schauble 2016) and it is essential that attention be paid to this issue by documenting whether the material displays spurious isotopic anomalies. Guideline #5: The choice of a reference material against which to report isotopic analyses should be a community-led effort and should be consistent. Once a standard has reached a certain level of acceptance (e.g., when more than 20 publications report isotopic compositions relative to that standard; if two reference materials have achieved this status, whichever has been more extensively used), that standard should be used in subsequent publications. If it is exhausted, a secondary reference material may be defined and an offset be applied to the isotopic analyses such that the original reference material can still be used to report isotopic compositions. For example, JMC Zn-Lyon is not readily available but future Zn isotopic analyses should still be reported relative to this

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standard even if new reference materials are used such as IRMM-3702 (Cloquet et al. 2006a; Ponzevera et al. 2006; Moeller et al. 2012) or NIST SRM-683 (Tanimizu et al. 2002; Chen et al. 2016). The secondary reference material used during the measurement should be specified when the δ-notation is defined. For example, one would write: “The Zn isotopic composition is expressed as, d66Zn = [(66Zn/64Zn)std / (66Zn/64Zn)JMC Lyon − 1] × 1000, where the isotopic analyses were measured relative to IRMM-3702 to which a systematic offset of − 0.29 was applied (Moeller et al. 2012) for conversion to the JMC Lyon d66Zn scale”. The exception to this guideline is that if the standard is proven to be isotopically heterogeneous as analytical capabilities improve, then a new reference material should be used to report isotopic compositions (see guideline # 1). Ideally, this new standard should have an isotopic composition that is similar to the original reference material used to define the d scale. Guideline #6: The reference materials should be widely available. This implies that labdefined and owned reference materials should be transferred to organizations that do not have a conflict of interest. The role of certification institutes such as IRMM or NIST is to characterize the reference materials that they distribute and ensure that they are available. In many instances, there has been a disconnect between the needs of the geochemical community and what these certification institutes can provide. This is probably due, in part, to the fact that non-traditional stable isotope geochemistry has grown at a rapid pace while preparation of certified materials can take a long time (Vogl and Pritzkow 2010). Part of the certification involves characterization of absolute isotopic abundances using gravimetrically prepared isotope mixtures. In nontraditional stable isotope geochemistry, knowing the absolute ratios is not particularly useful. To cope with the shortage or unavailability of isotopic reference materials, in-house isotopic standards have become the reference materials against which isotopic analyses are reported (DSM3 for Mg, Zn-Lyon for Zn, OL-Ti for Ti). An issue with this practice is the availability of those materials, and conflicts of interest may arise that are detrimental to the advancement of science. Investigator-controlled distribution systems do not work. Moving forward, some organizations/companies could take over that role and distribute (perhaps against a modest fee) the reference materials created by the community. Taking titanium as an example, the bar of pure Ti used to define the OL-Ti standard (Millet and Dauphas 2014; Millet et al. 2016) will be transferred to SARM (Nancy, France), where it will be available to end-users upon request. Another aspect regarding availability is the preparation of a large enough stock so that the reference material can be used for decades and the material remains stable in time. Aqueous solutions are appealing as they ensure homogeneity (Vogl and Pritzkow 2010) but the concentration is limited by solubility constraints and long-term stability may be an issue. Solids alleviate this issue and are more cost effective for the end user. Several isotopic systems are, or will be, in crisis if no action is taken to remediate the shortage of isotopic reference materials that can be used by newcomers to the field. There is no committee or working group overseeing the important issues related to standards and we take the opportunity of writing this chapter to propose the establishment of such a working group.

EMERGING ISOTOPE SYSTEMS The recent advances in instrumentation have made high-precision isotopic analysis possible for almost all elements on the periodic table. Besides those systems reviewed in individual chapters of this volume, there are numerous emerging systems that show great potential as briefly summarized for some of them below.

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Stable potassium isotope geochemistry Table 1. K (atomic number = 19) isotopes and typical natural abundance. Isotope

Abundance (%)

39

93.2581

40

0.0117

41

6.7302

K K K

Isotopic abundance data are from Berglund and Wieser (2011).

Potassium (K) is a volatile, lithophile, incompatible and fluid-mobile element (McDonough and Sun 1995). It is a major cation in both seawater and river water (Pilson 2013), and is well mixed in the ocean because of its long residence time of ~7 Myr (Li 1982). Potassium has three isotopes, 39K, 40K and 41K (Table 1). Among them, 40K is radioactive and exhibits a branched decay scheme to 40Ca and 40Ar, with a half-life of 1.25 × 109 years (Faure and Mensing 2005). The stable 39K and 41K isotopes have > 5% mass difference, which can potentially lead to large K isotope fractionations. Indeed, fractionation of K isotopes during chemical processes has been well known and was documented as early as 1938. Taylor and Urey (1938) found a 10% variation in 41K/39K when K was incompletely eluted by an aqueous solution from a zeolite ion exchange column, with 39K preferentially eluted from the exchange medium. This indicates that natural processes such as water-rock interactions could potentially fractionate K isotopes, and generate isotopically distinct reservoirs. Humayun and Clayton (1995a,b) found that both extraterrestrial samples (chondrites, eucrites, SNC meteorites, ureilites, and some lunar highland and mare igneous samples) and terrestrial samples (peridotites, basalts, granites, carbonatites, biotite schists and seawater) have similar K isotopic compositions within ± 0.5‰, despite variable levels of volatile element depletion among those bodies. This has been explained by vaporization under a high vapor pressure, as opposed to free evaporation. Indeed, under such conditions, evaporation would take place in an equilibrium rather than a kinetic regime (Richter et al. 2009 and references therein). Chondrules also revealed limited K isotope fractionation, again suggesting that evaporation took place under equilibrium conditions, presumably because chondrule melting and vaporization took place when the density of chondrules was high enough for a high partial pressure of K to build up around them (Alexander et al. 2000; Alexander and Grossman 2005). The advent of high-resolution mass spectrometry has made it possible to measure K isotopes with higher precision (from ± 0.1‰ to < ±0.05‰, 2SE by MC-ICPMS) (Morgan et al. 2014; Li et al. 2016; Wang and Jacobsen 2016a,b) and TIMS (Wielandt and Bizzarro 2011; Naumenko et al. 2013). Stable K isotopic compositions are reported in the δ notation:

 ( 41K/ 39K)  = δ41K(‰)  41 39 sample − 1 × 1000  ( K/ K)std  where the standard (std) is either NIST SRM 3141a (Li et al. 2016) or commercial ultrapure potassium nitrate (Wang and Jacobsen 2016a,b). There is a slight difference between these two standards based on the same geostandard and seawater data published from these two groups. Further studies are needed to better quantify the difference and to select one standard for reporting high-precision K isotopic data. To date, >1.4‰ variation in 41K/39K has been reported for terrestrial and extraterrestrial samples (Fig. 3). Morgan et al. (2014) found 0.4‰ K isotopic variation in a diverse range of whole rocks and mineral separates that formed at high temperatures, which is 10 times greater than the analytical uncertainty of  4‰ Cd isotopic variation in 114Cd/110Cd has been reported for terrestrial samples (Ripperger et al. 2007) and > 24‰ Cd isotopic variation in extraterrestrial samples (Wombacher et al. 2008). As reviewed in Rehkaemper et al. (2011), these large Cd isotopic variations mainly result from evaporation and condensation during both natural and anthropogenic processes as well as biological processes. Some ordinary chondrites, type 3 carbonaceous and most enstatite chondrites have highly fractionated Cd isotopic compositions, which reflect kinetic Cd isotope fractionation by volatilization and redistribution of Cd during open system thermal metamorphism on the parental bodies (Wombacher et al. 2003, 2008). The Cd produced during ore refineries has very different isotopic compositions relative to the natural Cd because of the evaporationdriven kinetic isotope fractionation during smelting of sulfide ores, which makes Cd isotopes a great tracer for anthropogenic sources of Cd into soils (Cloquet et al. 2006b; Shiel et al. 2010; Wen et al. 2015) and oceans (Shiel et al. 2012, 2013). Nonetheless, recent studies found significant Cd isotope fractionation during soil weathering, which may compromise the Cd isotopic signatures from pollution sources (Chrastny et al. 2015). Seawater displays the largest Cd isotopic variation in terrestrial samples because of the large Cd isotope fractionation during biological uptake and utilization of Cd in the seawater column. During Cd uptake into marine carbonates at the ocean surface (Boyle et al. 1976; Boyle 1988), light Cd isotopes are preferentially enriched in carbonates, which leads to a concomitant enrichment of heavy Cd isotopes (> +2‰) in the surface water. The Cd isotopic composition of deep seawater (>1000 m) seems homogenous, with d114Cd = ~+0.3‰ (relative to NIST SRM 3108; Conway and John 2015, and the references therein). In addition to the vertical heterogeneity, surface seawaters from different oceans also display a large isotopic variation, which reflects both isotope fractionations and mixing among the oceans. Therefore, Cd isotope geochemistry can be used to trace not only global Cd cycle but also changes in marine biological activity in the past (Lacan et al. 2006; Ripperger et al. 2007; Schmitt et al. 2009b; Abouchami et al. 2011; Xue et al. 2013; Abouchami et al. 2014; Conway and John 2015; Georgiev et al. 2015).

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Tin isotope geochemistry Table 7. Sn (atomic number = 50) isotopes and typical natural abundance. Isotope

Abundance (%)

112

0.97

114

0.66

Sn Sn

115

0.34

116

14.54

Sn Sn

117

7.68

118

24.22

119

8.59

120

32.58

122

4.63

124

5.79

Sn Sn Sn Sn Sn Sn

Isotopic abundance data are from Berglund and Wieser (2011).

Tin (Sn) is a chalcophile and highly volatile trace element with two oxidation states (Sn2+ and Sn4+). Natural Sn is mainly associated with sulfide minerals though Sn oxide (cassiterite, SnO2) is also widespread. Tin has 10 stable isotopes with a large relative mass difference (> 10%) (Table 7). Sn isotope fractionation was considered negligible until 2002 when Clayton et al. (2002) developed a method for high-precision isotopic analysis of Sn based on MC-ICPMS. Since then, tin isotopes have been widely applied in archaeology to trace Sn provenance of artifacts. This is because Sn was a vital commodity in the past and Sn from different ore deposits from different locations has different isotopic compositions (Haustein et al. 2010; Balliana et al. 2013; Yamazaki et al. 2013, 2014; Mason et al. 2016). Nonetheless, the application of Sn isotopes in geochemistry is still limited. A remarkable feature of Sn is that like Fe, it possesses a Mössbauer isotope 119Sn, so it is possible to predict its equilibrium fractionation factor using the synchrotron technique of Nuclear Resonant Inelastic X-ray Scattering (NRIXS) or conventional Mössbauer spectroscopy. Polyakov et al. (2005) used these techniques to predict Sn isotope fractionation for different phases and found a large control of its different oxidation states. Malinovskiy et al. (2009) showed that UV irradiation of methyltin can cause large mass-dependent and massindependent Sn isotope fractionations. Moynier et al. (2009) observed mass-independent Sn isotope fractionation in chemical exchange reactions by using dicyclohexano-18-crown-6, which they argued are due to nuclear field shift effects.

Antimony isotope geochemistry Table 8. Sb (atomic number = 51) isotopes and typical natural abundance. Isotope

Abundance (%)

121

Sb

57.21

123

Sb

42.79

Isotopic abundance data are from Berglund and Wieser (2011).

Antimony (Sb) is a siderophile, moderately incompatible trace element in the Earth (McDonough and Sun 1995), with multiple oxidation states (dominated by Sb3+, Sb5+ and Sb3−). It is toxic, with Sb3+ more toxic than Sb5+. Antimony has two stable isotopes (Table 8). At present, the literature on high-precision Sb isotopic data is still limited. Rouxel et al. (2003) characterized the Sb isotopic compositions of seawater, silicate rocks, various environmental samples, deep-sea sediments and hydrothermal sulfides. The continental and oceanic crustal rocks have relatively restricted and lighter Sb isotopic composition than seawater, which has a homogenous Sb isotopic composition. The hydrothermal sulfides display the largest (up

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to 1.8‰) Sb isotopic variation. The large Sb isotopic variation reflects both heterogeneous Sb sources and isotope fractionation during redox changes in aqueous solutions at low temperatures (Rouxel et al. 2003). Recently, Resongles et al. (2015) have found that two rivers in France, which drained different mining sites, have distinct Sb isotopic compositions, reflecting Sb isotope fractionation during complexed Sb transfer from rocks, mine wastes and sediments to the river water. Overall, Sb isotope systematics may be a useful tool for tracing redox processes, pollution sources and biogeochemical processes in riverine and oceanic systems (Rouxel et al. 2003; Resongles et al. 2015).

Stable tellurium isotope geochemistry Table 9. Te (atomic number = 52) isotopes and typical natural abundance, Isotope

Abundance (%)

120

0.09

122

2.55

123

0.89

124

4.74

Te Te Te Te

125

7.07

126

18.84

128

31.74

130

34.08

Te Te Te Te

Isotopic abundance data are from Berglund and Wieser (2011).

Tellurium (Te) belongs to the same group as S and Se, and is a volatile, chalcophile trace element with multiple oxidation states (Te6+, Te4+, Te2+, and Te2−). Tellurium has eight isotopes with > 8% relative mass difference (Table 9). Among them, 126Te is the decay product of 126Sn, with a half-life of 0.2345 million years (Oberli et al. 1999). In addition, the eight Te isotopes were produced by r-, s-, and p- processes during stellar nucleosynthesis. Hence, Te isotopes have been mainly used in cosmochemistry for searching the shortlived 126Sn in order to use the 126Sn–126Te chronometer to understand early solar system processes, or for searching the nucleosynthetic anomaly in order to understand the solar nebular processes (Fehr et al. 2004, 2005, 2006, 2009; Moynier et al. 2009; Fukami and Yokoyama 2014). To our knowledge, there is only one study that focuses on stable Te isotope geochemistry. Fornadel et al. (2014) developed a method for high-precision (~±0.10‰, 2SD for 130Te/125Te) analysis of Te isotopes in tellurides and native tellurium from various ore deposits and documented an over 1.6‰ Te isotopic variation. Though the processes responsible for the large fractionation is unknown, Te isotope geochemistry, could become an excellent complement to S and Se isotope geochemistry.

Barium isotope geochemistry Barium (Ba) is an alkaline earth element with seven stable isotopes (Table 10). Most of the work in Ba stable isotope geochemistry has focused on Ba isotope anomalies in extraterrestrial materials (Eugster et al. 1969; McCulloch and Wasserburg 1978; Hidaka et al. 2003; Savina et al. 2003; Ranen and Jacobsen 2006; Andreasen and Sharma 2007; Carlson et al. 2007; Hidaka and Yoneda 2011). The past several years have seen a rapidly growing literature on Ba isotopes in terrestrial materials. The Ba isotopic compositions are most often reported using isotopes 137Ba and 134Ba: 137 134  ( Ba/ Ba)  = δ137 Ba(‰)  137 134 sample − 1 × 1000   ( Ba/ Ba)std

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where d137Ba has been reported relative to one of three different standards in the recent literature: (1) a Ba(NO3)2 solution from Fluka Aldrich (Von Allmen et al. 2010; Miyasaki et al. 2014; Pretet et al. 2016), (2) a BaCO3 solution (IAEA-CO-9), and (3) a Ba(NO3)2 solution from the National Institute of Standards and Technology (SRM3104a; Nan et al. 2015, Horner et al. 2015). Table 10. Ba (atomic number = 56) isotopes and typical natural abundance. Isotope

Abundance (%)

130

Ba

0.106

132

Ba

0.101

134

Ba

2.417

135

Ba

6.592

136

Ba

7.854

137

Ba

11.232

138

Ba

71.698

Isotopic abundance data are from Berglund and Wieser (2011). 137 134

The total natural variation in Ba/ Ba of natural terrestrial materials is ~ 1.5‰ (Fig. 5). Although this range is quite restricted (owing to the small relative mass difference between Ba isotopes), modern MC-ICPMS methods are able to yield a reproducibility of about ±0.03‰ to ±0.05‰ (Miyazaki et al. 2014; Nan et al. 2015), which is more than an order of magnitude smaller than the natural variations. The most recent applied studies have focused primarily on the Ba cycle in seawater because the concentration of Ba closely tracks the concentration of dissolved Si(OH)4 and other nutrients (von Allmen et al. 2010; Horner et al. 2015; Pretet et al. 2015; Cao et al. 2016). The thinking behind these studies is that Ba isotopes coupled with Ba concentration measurements may be useful to probe nutrient cycling, biologic productivity, and water mass mixing (Horner et al. 2015; Cao et al. 2016). Ba enters the oceans from rivers with a d137Ba of about 0 to +0.3‰ (Fig. 6). From there, precipitation of BaSO4 in organic microenvironments or adsorption of Ba onto organic particulates leads to light isotope enrichment in the solid phase, and consequently, seawater becomes isotopically heavy with respect to Ba (Fig. 6; Horner et al. 2015; Cao et al. 2016). This general process leads to a strong correlation between Ba2+  concentrations and d137Ba, and imparts a distinct chemical and isotopic signature on different water masses. Hence, coupled Ba–d137Ba systematics can be used to trace the mixing of different water masses (Horner et al. 2015; Cao et al. 2016). An open question is whether other factors such as hydrothermal activity, submarine groundwater discharge, or atmospheric inputs, are important within the oceanic Ba cycle or simply have effects that cancel out (Cao et al. 2016). Like particulate matter, corals are isotopically lighter than seawater (Fig. 6; Pretet et al. 2015). Most of the corals measured so far were analyzed without information regarding the isotopic composition of the host seawater, but the expectation from laboratory cultures is that coral is lighter than seawater by 0.01 to 0.26 ± 0.14‰ (Pretet et al. 2015). To aid the interpretation of d137Ba in nature, several studies have investigated isotope fractionation by diffusion (Van Zuilen et al. 2016) and between experimentally grown inorganic minerals and their host solution. Von Allmen et al. (2010) grew BaCO3 and BaSO4 from aqueous solutions at 21 and 80 ºC. Like many previous experiments using other isotopic systems, they observed that the solid phase was isotopically light relative to the aqueous solution. For BaCO3, the fractionation ranges from 0.1 (fast growth) to 0.3‰ (slow growth). An interesting result for both BaCO3 and BaSO4 is that the fractionations are insensitive to temperature between 21 and 80 ºC. Mavromatis et al. (2016) precipitated BaCO3 at 25 ºC and documented light isotope enrichment in the solid phase by 0.07 ± 0.04‰, a value that is in

Teng, Dauphas, & Watkins

18

Fluka (-0.032)

SRM3104a (-0.017)

2 s.d. natural barite diagenetic barite terrestrial gangue igneous rocks rivers ocean particulates corals seawater −1.0

−0.5

0

0.5

1.0

δ Ba (‰) 137

Figure 6. Natural variations in the Ba isotopic composition of terrestrial materials relative to the standard IAEA-CO-9. The vertical line and shaded area represent the standard value with d137Ba = 0.00 ± 0.02 (2SD). A different choice of standard, shown by the two arrows, would not shift δ values significantly (Miyazaki et al. 2014; Nan et al. 2015). Data are from the literature (Von Allmen et al. 2010; Miyazaki et al. 2014; Horner et al. 2015; Nan et al. 2015; Cao et al. 2016; Pretet et al. 2016).

good agreement with the “fast growth” experiments of Von Allmen et al. (2010). Mavromatis et al. (2016) went a step further and monitored the Ba isotope composition of the aqueous solution over time after chemical (but not isotopic) equilibrium between BaCO3 and fluid was established. After about 6–8 days, the solution had evolved isotopically to a composition that was indistinguishable from the solid phase, indicating a BaCO3–fluid equilibrium fractionation factor of ~0‰. Such a small equilibrium effect is consistent with similarities in the Ba–O bond length of witherite (2.80 Å) compared to aqueous Ba (2.79 Å) (Mavromatis et al. 2016). One other experimental study by Böttcher et al. (2012) showed that BaMn(CO3)2 grown at 21 ºC yields a similar light isotope enrichment in the solid phase of 0.11 ± 0.06‰.

Stable neodymium isotope geochemistry Neodymium (Nd) is a rare earth element, with seven isotopes (Table 11). Of them, 142Nd is the decay product of 146Sm, with a half-life of 68 million years (Kinoshita et al. 2012) and 143 Nd is the decay product of 147Sm, with a half-life of 106 billion years (Faure and Mensing 2005). The 147Sm–143Nd and 146Sm–142Nd have been extensively used as chronometers and fingerprints of mantle source regions (DePaolo 1988; Boyet and Carlson 2005; Faure and Mensing 2005). Because of the small relative mass difference, the stable Nd isotope ratios are often treated as invariable, with 146Nd/144Nd = 0.7219. This value has been used to calibrate instrumental fractionations for high-precision Nd isotopic analyses.

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Table 11. Nd (atomic number = 60) isotopes and typical natural abundance. Isotope

Abundance (%)

142

Nd

27.152

143

Nd

12.174

144

Nd

23.798

145

Nd

8.293

146

Nd

17.189

148

Nd

5.756

150

Nd

5.638

Isotopic abundance data are from Berglund and Wieser (2011).

With the developments in high precision mass spectrometry using MC-ICPMS and TIMS, significant stable isotope fractionation has been found for Nd in natural samples. The large fractionations have important implications for the measurement and application of radiogenic Nd isotopes, as well as great potential to trace various geological processes. The stable Nd isotopic compositions are often reported in the ε notation: x 144  ( Nd/ Nd)sample  = ε x Nd  x − 1 × 10000 144   ( Nd/ Nd)std

where the standard (std) is the JNdi−1 and x = mass 145, 146, 148 and 150. When x = 142 and 143, the e142Nd and e143Nd represent the radiogenic Nd isotopic compositions. The literature on stable Nd isotope geochemistry is still very limited. Nonetheless, the data so far reveal significant Nd isotope fractionation (Fig. 7). Wakaki and Tanaka (2012) reported high-

Figure 7. Natural variations in the stable Nd isotopic composition of terrestrial materials relative to the standard JNdi−1. The La Jolla Nd has different d146Nd value than the JNdi−1. Data are from Wakaki and Tanaka (2012), and Ma et al. (2013b).

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Teng, Dauphas, & Watkins

precision (±0.3 for e146Nd, 2SD) stable Nd isotope analysis by using DS-TIMS method and found large mass-dependent Nd isotope fractionation during column chemistry, with e146Nd varying from +4.60 at the early stage of elution to −9.44 at the late stage of elution. They also found significant Nd isotopic variation in commercial high-purity Nd reagents (e146Nd = −2.36 to + 0.23). In particular, the La Jolla Nd (e146Nd = −1.97 ± 0.23) has different stable Nd isotopic composition than JNdi−1. Ma et al. (2013b) developed a method of using MC-ICPMS to measure stable Nd isotopes and achieved similar precision (better than ±0.4 for e146Nd, 2SD) to DS-TIMS. They found a large Nd isotopic variation in 15 rocks and polymetallic nodules (mainly geostandards), with e146Nd ranging from −2.65 in a stream sediment (JSD-1) to +2.12 in a granodiorite (JG-1a). The in-house Nd standard (Nd-GIG) has the heaviest Nd isotopic composition, with e146Nd= + 2.25. Overall, relative to basalts, river and marine sediments tend to have light Nd isotopic compositions while granites and granodiorites have heavy Nd isotopic compositions.

CONCLUSIONS Our knowledge of the behavior and utility of non-traditional stable isotopes has expanded greatly since the publication of the RIMG volume 55: Geochemistry of Non-Traditional Stable Isotopes in 2004. A testament to the rapid progress in this field is that many of the stable isotope systems that were in their infancy in 2004 have a dedicated chapter in this volume. We anticipate that the isotope systems summarized in this introductory chapter or not even mentioned (e.g., Ga, Br and Zr) will warrant their own dedicated chapter in the not too distant future.

ACKNOWLEDGEMENTS FZT was supported by NSF (EAR-0838227 and EAR-1340160). ND was supported by NASA (NNX14AK09G, OJ-30381-0036A and NNX15AJ25G) and NSF (EAR144495 and EAR150259). JMW was supported by the NSF (EAR-1249404 and EAR-1050000). FZT thanks Jinlong Ma, Lie-Meng Chen, Yong-Sheng He, Zhuang Ruan, Kwan-Nang Pang, Wang-Ye Li, Yan Hu, Ben-Xun Su, Yang Sun, Shui-Jiong Wang for their helpful comments and discussion.

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Stevenson EI, Hermoso M, Rickaby REM, Tyler JJ, Minoletti F, Parkinson IJ, Mokadem F, Burton KW (2014) Controls on stable strontium isotope fractionation in coccolithophores with implications for the marine Sr cycle. Geochim Cosmochim Acta 128:225–235 Stevenson EI, Aciego SM, Chutcharavan P, Parkinson IJ, Burton KW, Blakowski MA, Arendt CA (2016) Insights into combined radiogenic and stable strontium isotopes as tracers for weathering processes in subglacial environments. Chem Geol 429:33–43 Stüeken EE (2017) Selenium isotopes as a biogeochemical proxy in deep time. Rev Mineral Geochem 82:657–682 Tanimizu M, Asada Y, Hirata T (2002) Absolute isotopic composition and atomic weight of commercial zinc using inductively coupled plasma mass spectrometry. Anal Chem 74:5814–5819 Taylor TI, Urey HC (1938) Fractionation of the Li and K isotopes by chemical exchange with zeolites. J Chem Phys 6:429–438 Teng F-Z (2017) Magnesium isotope geochemistry. Magnesium isotope geochemistry. Rev Mineral Geochem 82:219–287 Trinquier A, Elliott T, Ulfbeck D, Coath C, Krot AN, Bizzarro M (2009) Origin of nucleosynthetic isotope heteorogeneity in the solar protoplanetary disk. Science 324:374–376 Valley JW, Cole DRE (Eds.) (2001) Stable Isotope Geochemistry. Rev Mineral Geochem 43, Mineralogical Society of America and the Geochemical Society, Washington DC Valley JW, Taylor HP, O’Neil JR (1986) Stable Isotopes in High-Temperature Geological Processes. Rev Mineral 16. The Mineralogical Society of America, Washington DC van Zuilen K, Müller T, Nägler TF, Dietzel M, Küsters T (2016) Experimental determination of barium isotope fractionation during diffusion and adsorption processes at low temperatures. Geochim Cosmochim Acta 186:226–24 Vogl J, Pritzkow W (2010) Isotope reference materials for present and future isotope research. J Anal At Spectrom 25:923–932 Vollstaedt H, Eisenhauer A, Wallmann K, Bohm F, Fietzke J, Liebetrau V, Krabbenhoft A, Farkas J, Tomasovych A, Raddatz J, Veizer J (2014) The Phanerozoic δ88/86Sr record of seawater: New constraints on past changes in oceanic carbonate fluxes. Geochim Cosmochim Acta 128:249–265 Von Allmen K, Böttcher ME, Samankassou E, Nägler TF (2010) Barium isotope fractionation in the global barium cycle: First evidence from barium minerals and precipitation experiments. Chem Geol 277:70–7 Waight T, Baker J, Willigers B (2002) Rb isotope dilution analyses by MC-ICPMS using Zr to correct for mass fractionation: towards improved Rb-Sr geochronology? Chem Geol 186:99–116 Wakaki S, Tanaka T (2012) Stable isotope analysis of Nd by double spike thermal ionization mass spectrometry. Inter J Mass Spectrom 323:45–54 Wang K, Jacobsen SB (2016a) An estimate of the Bulk Silicate Earth potassium isotopic composition based on MC-ICPMS measurements of basalts. Geochim Cosmochim Acta 178:223–232 Wang K, Jacobsen SB (2016b) Potassium isotopic evidence for a high-energy giant impact origin of the Moon. Nature 538:487–490 Wei GJ, Ma JL, Liu Y, Xie LH, Lu WJ, Deng WF, Ren ZY, Zeng T, Yang YH (2013) Seasonal changes in the radiogenic and stable strontium isotopic composition of Xijiang River water: Implications for chemical weathering. Chem Geol 343:67–75 Wen HJ, Zhang YX, Cloquet C, Zhu CW, Fan HF, Luo CG (2015) Tracing sources of pollution in soils from the Jinding Pb-Zn mining district in China using cadmium and lead isotopes. Appl Geochem 52:147–154 Widanagamage IH, Schauble EA, Scher HD, Griffith EM (2014) Stable strontium isotope fractionation in synthetic barite. Geochim Cosmochim Acta 147:58–75 Widanagamage IH, Griffith EM, Singer DM, Scher HD, Buckley WP, Senko JM (2015) Controls on stable Srisotope fractionation in continental barite. Chem Geol 411:215–227 Wielandt D, Bizzarro M (2011) A TIMS-based method for the high precision measurements of the three-isotope potassium composition of small samples. J Anal At Spectrom 26:366–377 Wombacher F, Rehkaemper M, Mezger K, Munker C (2003) Stable isotope compositions of cadmium in geological materials and meteorites determined by multiple-collector ICPMS Geochim Cosmochim Acta 67:4639–4654 Wombacher F, Rehkaemper M (2004) Problems and suggestions concerning the notation of cadmium stable isotope compositions and the use of reference materials. Geostand Geoanal Res 28:173–178 Wombacher F, Rehkamper M, Mezger K, Bischoff A, Munker C (2008) Cadmium stable isotope cosmochemistry. Geochim Cosmochim Acta 72:646–667 Wu F, Qin T, Li X, Liu Y, Huang JH, Wu Z and Huang F (2015) First-principles investigation of vanadium isotope fractionation in solution and during adsorption. Earth Planet Sci Lett 426:216–224 Wu F, Qi Y, Yu H, Tian S, Hou Z and Huang F (2016) Vanadium isotope measurement by MC-ICP-MS Chem Geol 421:17–25 Xue ZC, Rehkamper M, Horner TJ, Abouchami W, Middag R, van de Flierdt T, de Baar HJW (2013) Cadmium isotope variations in the Southern Ocean. Earth Planet Sci Lett 382:161–172

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Reviews in Mineralogy & Geochemistry Vol. 82 pp. 27-63, 2017 Copyright © Mineralogical Society of America

Equilibrium Fractionation of Non-traditional Isotopes: a Molecular Modeling Perspective Marc Blanchard, Etienne Balan Institut de Minéralogie, de Physique des Matériaux, et de Cosmochimie (IMPMC) Sorbonne Universités UPMC Univ Paris 06 UMR CNRS 7590 Muséum National d’Histoire Naturelle UMR IRD 206 4 place Jussieu F-75005 Paris France [email protected]; [email protected]

Edwin A. Schauble Department of Earth Planetary and Space Sciences University of California, Los Angeles Los Angeles California 90095-1567 U.S.A. [email protected]

INTRODUCTION The isotopic compositions of natural materials are determined by their parent reservoirs, on the one hand, and by fractionation mechanisms, on the other hand. Under the right conditions, fractionation represents isotope partitioning at thermodynamic equilibrium. In this case, the isotopic equilibrium constant depends on temperature, and reflects the slight change of free energy between two phases when they contain different isotopes of the same chemical element. The practical foundation of the theory of mass-dependent stable isotope fractionation dates back to the mid-twentieth century, when Bigeleisen and Mayer (1947) and Urey (1947) proposed a formalism that takes advantage of the Teller–Redlich product rule (Redlich 1935) to simplify the estimation of equilibrium isotope fractionations. In this chapter, we first give a brief introduction to this isotope fractionation theory. We see in particular how the various expressions of the fractionation factors are derived from the thermodynamic properties of harmonically vibrating molecules, a surprisingly effective mathematical approximation to real molecular behavior. The central input data of these expressions are vibrational frequencies, but an approximate formula that requires only force constants acting on the element of interest can be applied to many non-traditional isotopic systems, especially at elevated temperatures. This force-constant based approach can be particularly convenient to use in concert with first-principles electronic structure models of vibrating crystal structures and aqueous solutions. Collectively, these expressions allow us to discuss the crystal chemical parameters governing the equilibrium stable isotope fractionation. 1529-6466/17/0082-0002$05.00

http://dx.doi.org/10.2138/rmg.2017.82.2

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Since the previous volume of Reviews in Mineralogy and Geochemistry dedicated to non-traditional stable isotopes, the number of first-principles molecular modeling studies applied to geosciences in general and to isotopic fractionation in particular, has significantly increased. After a concise introduction to computational methods based on quantum mechanics, we will focus on the modeling of isotopic properties in liquids, which represents a bigger methodological challenge than small molecules in gas phase, or even minerals. Our ability to produce reliable theoretical mineral–solution isotopic fractionation factors is essential for many geosciences problems. The main modeling approaches used in recent studies of fractionation in liquids are molecular cluster models and molecular dynamics with periodic boundary conditions. Their relative advantages and drawbacks will be discussed. So far, the vast majority of theoretical studies applied to isotopic fractionation have been based on the harmonic approximation; in most cases anharmonic effects will be smaller than uncertainties associated with other imperfections in the models (especially in calculated vibrational frequencies), but in some cases (e.g., liquid phases with light elements) it will be important to be able to go beyond the harmonic approximation. More sophisticated methods, such as thermodynamic integration coupled to path integral molecular dynamics, can account for anharmonic effects as well as quantum nuclear effects. We will introduce the basic concepts of this technique and will give some examples of their application. Among the non-traditional isotopes, the iron isotope system has probably developed the richest and most methodologically varied theoretical literature. This is partly due to the fact that isotope fractionation factors of Mössbauer-active elements (including iron, via 57Fe) can be independently determined using Mössbauer spectroscopy and nuclear resonant inelastic X-ray scattering, which are closely related techniques that probe the vibrational properties of the target element. Expressions used to derive fractionation factors from these spectroscopic techniques are introduced, the accuracy of each method will be discussed, and the results are compared with first-principles calculations. The discovery of mass-independent isotope fractionation of non-traditional stable isotope systems including Hg, Tl, and U over the past decade has expanded the scope of “stable” isotope geochemistry to include a long-lived radioactive element and almost the whole range of naturally occurring atomic numbers. It has also created a need for theoretical studies of new fractionation mechanisms. Nuclear field shift effects, first proposed to explain laboratory isotope enrichment experiments, including uranium, are now thought to play an important role in driving natural fractionation in uranium and thallium, and a secondary role in mercury isotope geochemistry. Large photochemically induced mass-independent fractionation effects in the mercury isotope system are yet to be explained beyond a qualitative level, and remain an important challenge for isotope fractionation theory. In light isotope systems (particularly 16 O–17O–18O) it is now possible to measure variability of mass dependence for different types of fractionation, ranging from equilibrium partitioning to kinetic fractionation. The potential for using variations in mass dependence to identify the types of fractionation affecting nontraditional elements is also a topic of emerging interest for theoretical studies.

THEORETICAL FRAMEWORK Equilibrium fractionation theory This section is largely inspired by the articles of Bigeleisen and Mayer (1947), and Ishida (2002). Let’s consider an isotopic exchange reaction between two molecules A and B, involving a single atomic position:

AX ' + BX  AX + BX '

(1)

Equilibrium Fractionation: A Molecular Modeling Perspective

29

The prime symbol refers to the light isotope of the element X. As with any chemical reaction, the equilibrium constant, Keq, can be determined from the free energies of the reactants and products. Isotopic exchange reactions do not, in general, involve significant pressure-volume work because the number of molecules on both sides of the reaction is the same, and because isotope substitution has a negligible effect on the molar volumes of the phases under normal conditions. The above assumption is not true for a complete substitution of hydrogen by deuterium, for instance, or for certain solid–gas equilibria (e.g., Jancso et al. 1993; Horita et al. 2002). Under these general conditions, the standard Gibbs free energy of the exchange reaction can be related to the difference in the Helmholtz free energy of the pure isotopomers (AX', AX, BX' and BX): = ∆F F ( AX ) + F ( BX ') − F ( AX ') − F ( BX ) The Helmholtz free energy is related to the molecular partition function, Q by:

F=

− N a kT ln ( Q / N a )

where k is Boltzmann’s constant, Na the Avogadro number, and T is the absolute temperature. It is thus possible to express the equilibrium constant of the exchange reaction as:

K eq =

Q( AX ) × Q( BX ') Q( AX ') × Q( BX )

(2)

The molecular partition function is given by the following expression: Q = ∑ exp( − En / kT ) n

where the sum spans all the quantum states of the molecule, referred to by their index n and their corresponding energies En. A classical partition function, Qcl can be obtained by integration over continuous momenta and position variables that relate to the kinetic energy and potential energy of the molecule, respectively. Its expression is also a function of the symmetry number of the molecule (i.e. the number of equivalent ways to orient a molecule in space); see Equation (5) of Bigeleisen and Mayer (1947). Importantly, atomic masses are only involved in the definition of the kinetic energy term; whereas the configurational integral, obtained from the potential energy of the system, is assumed to be mass-independent. Considering the ratio of the partition functions of the two isotopically substituted molecules, the configurational integrals and the contribution arising for atoms other than the exchanged isotopes cancel out, leading to the following expression: s'  m  Q   =   Q '  cl s  m ' 

3

2

where s is the symmetry number and m, the atomic mass of isotopes. By inserting this expression into Equation (2), the atomic masses cancel out and one obtains the classical value of the equilibrium constant:

( K eq )cl =

s AX ' sBX s AX sBX '

(3)

This ratio of symmetry numbers will not lead to an isotopic fractionation as it merely represents the relative probabilities of forming symmetric and antisymmetric molecules. This corresponds to a perfectly random distribution of isotopes; a situation found at T = ∞. This demonstrates that isotopic fractionation is a purely quantum effect that cannot be explained by classical statistical mechanics.

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The harmonic quantum molecular partition function of a molecule in gas phase, can be written as a product of translational, rotational, vibrational and electronic partition functions: Qqm = ( QT )qm × ( QR )qm × ( QV )qm × ( QE )qm

The electronic structure of a molecule is usually assumed to be isotope-independent, so the electronic term is neglected. In some cases, isotope effects will have measurable electronic contributions: isotope mass shift, nuclear field shift effect and nuclear spin effect (e.g. Bigeleisen 1996). Fractionations caused by the nuclear field shift and spin effects will be discussed in a later section. The mass shift arises from the coupling of the motion of the nuclei and the electrons. Mass shift can be important in reaction involving hydrides, i.e. up to a small percentage of H–D fractionation factors (e.g. Kleinman and Wolfsberg 1973), but becomes quickly negligible for heavier atoms since it scales with δM/M2. Partition functions for translational and rotational motions are formally quantum mechanical, and sensitive to isotope substitution, but in practice the quanta for both types of motion are so small and closely spaced for most molecules that they do not deviate significantly from their classical equivalents at temperatures relevant to geochemistry (see Schauble 2004, for additional details). This will hold in all cases except hydrogen, where a more sophisticated treatment of rotation may be needed at low temperatures. The classical expressions are not given here since most of their parameters will cancel out in the next step, but they can be found in literature (e.g., Richet et al. 1977; Schauble 2004; Liu et al. 2010). As a result, it is generally the case that only vibrational motions need a quantum-mechanical treatment, and it is vibrational energy that plays the central role in controlling the distribution of isotopes between two phases in thermodynamic equilibrium. The harmonic vibrational partition function is defined by: 3 N −6

(QV )qm = ∏ i =1

exp ( − hν i / 2kT )

1 − exp ( − hν i / kT )

where h is Planck’s constant and νi is the frequency of the vibrational mode i. A molecule with N atoms will have 3N − 6 vibrational degrees of freedom (in addition to 3 rotational and 3 translational degrees of freedom) while a linear molecule will have 3N − 5 vibrational degrees of freedom. As classical contributions to the partition functions only play a bookkeeping role in the isotopic fractionation, it is useful to define a reduced partition function by ratioing the quantum partition function to its classical counterpart (Qqm/Qcl). The ratio of reduced partition function, commonly referred to as β, can be written as: Q ( AX ) Qcl ( AX ) β AX = qm Qqm ( AX ' ) Qcl ( AX ' )

(4)

The equilibrium isotope fractionation factor of the exchange reaction (1), i.e. αAX-BX, can thus be expressed as a function of the reduced partition function ratios and is related to the equilibrium constant through the following relation (using Eqns. (2) and (3)): β AX Qqm ( AX ) × Qqm ( BX ') Qcl ( AX ') × Qcl ( BX ) = × β BX Qqm ( AX ') × Qqm ( BX ) Qcl ( AX ) × Qcl ( BX ')

a AX= - BX = =

(K eq )qm (K eq )cl sAX' sBX × × (K eq )qm sAX sBX'

Equilibrium Fractionation: A Molecular Modeling Perspective

31

The above relation shows that, when (Keq)qm is equal to (Keq)cl, there will be no isotopic fractionation (i.e. αAX-BX = 1). This situation occurs, for instance, at very high temperatures when isotopes are randomly distributed. Conversions between fractionation factors and equilibrium constant, written here for the simple isotopic exchange reaction (1), can be more complicated, depending on molecular stoichiometry (Schauble 2004; Liu et al. 2010). By analogy with the isotopic fractionation factor α, we can also see the reduced partition function ratio βAX as the isotopic fractionation factor between the substance AX and an ideal atomic gas of X. This formulation is a convenient way to tabulate the theoretical fractionations with a simple point of comparison. Fractionations are typically very small, on the order of parts per thousand for non-traditional stable isotopes, so it is common to use the notation 1000 ln α or 1000 ln β expressing the result in permil (‰). The β-factor being the central quantity of theoretical studies, we report below the expressions that apply to situations commonly encountered (i.e. molecules or condensed phases, complete or site by site isotopic substitution). By inserting the expressions of the quantum and classical partition functions into Equation (4), we obtain the following expression for a molecule in gas phase having only one exchangeable atom: 1

2  3 N − 6 exp ( − hν i 2kT ) 1-exp ( − hν 'i kT )  (5)  m '  2  M  2  IxIyIz  β= AX  m  ×  M '  ×  I ' I ' I '  ×  ∏ 1 − exp − hν kT × exp − hν ' 2kT  ( i ) ( i )      x y z  i =1 where M is the mass of molecule AX, M' is the mass of molecule AX', and Ix, I'x, etc. are the moments of inertia along each cartesian axis. 3

3

Alternatively, for a molecule having n exchangeable atoms, the βAX can be determined from the βi related to a specific atomic site i: 1 n 1 n Qqm ( AX 'n −1 Xi )  m '  = βi ∑ ∑ Q ( AX ' ) ×  m  n i 1= ni1 = n qm

3/ 2

(6)

β AX =

In this “site by site” approach, Qqm(AX'n-1Xi) corresponds to the partition function of the molecule having the atom X' on the site i substituted with X while Qqm(AX'n) represents the partition function of the molecule with no substituted atoms. If we further use the Teller–Redlich product rule (e.g., Redlich 1935; Wilson et al. 1955): 1

3

 IxIyIz  2  M  2  m '    ×   × m   M'    I ' x I ' y I 'z 

3N

2

3 N −6

×∏ i =1

ν 'i = 1 νi

then Equation (5) transforms into a more general expression applicable to any molecule in gas phase undergoing a complete isotopic substitution (i.e. all X' atoms are substituted with X): 3 N −6 ν exp( − hν i / 2kT ) 1 − exp( − hν 'i / kT )  = β AX  ∏ i × ×   i =1 ν 'i 1 − exp( − hν i / kT ) exp( − hν 'i / 2kT ) 

1

n

(7)

This more convenient form involving only the vibrational frequencies before and after full isotope substitution, assumes that: (i) the free energy change associated to the isotopic substitution of an atom X does not depend on the isotopic nature of the surrounding X atoms, and (ii) the β-factor of each atomic site weakly depends on the site. These assumptions are generally valid except in some specific cases, like for instance, when deuterium is substituted for hydrogen in a water molecule. Crystalline materials differ from gaseous molecules by their spatial extension involving the presence of long-range interactions. This implies that their vibrational spectra do not exhibit a finite number of vibrational frequencies but rather correspond to a continuum. In

Blanchard, Balan & Schauble

32

crystals, a vibrational mode is defined by a frequency of vibration, the atomic displacement pattern in a given cell and a wave-vector q that describes the phase relation of the atomic displacements in the other cells of the crystal. The wave-vector is defined in the reciprocal space and belongs to the first Brillouin zone. The vibrational frequency thus depends on the wave-vector. It is possible to build dispersion curves by reporting the frequency along specific directions in the reciprocal space, and vibrational density of states by integration over the whole Brillouin zone. A more detailed description of the crystal vibrational properties applied to isotope fractionation can be found in Young et al. (2015). This feature of crystals can be taken into account by modifying the partition function. The energy differences associated with translation motions cancel at equilibrium and the rotational term disappears in crystals. The partition function, in the harmonic approximation, is thus defined by:  3N exp ( − hν q ,i / 2kT )  =  ∏∏   i =1 {q} 1 − exp ( − hν q ,i / kT )   

1/ N q

Qqm

(8)

where νq,i is now the frequency of the vibrational mode i, along the wave-vector q. N corresponds to the number of atom in the crystal unit cell. The second product is performed on a uniform grid of Nq q-vectors in the Brillouin zone. In practice, the number of frequencies used is still finite but beyond a sufficiently large number of q-vectors, results are properly converged. By combining Equations (4) and (8), we obtain the general expression of the reduced partition function ratio (i.e. β-factor) for a crystal undergoing a complete isotopic substitution (i.e. all n atoms X' of the unit-cell are substituted with X): exp ( − hν q ,i / 2kT ) 1 − exp ( − hν′q ,i / kT )   m '  2  3N β= ×  AX  m  × ∏∏    i =1 {q} 1 − exp ( − hν q ,i / kT ) exp ( − hν′q ,i / 2kT )  3

1

n Nq

(9)

Alternatively, Equation (6) for a “site by site” isotopic substitution is still valid for crystals. The rotational and translational terms also disappear from the Teller–Redlich product rule, yielding the high-temperature product rule of Kieffer (1982): 3

ν 'q , i   m '  2  3N  m  ×  ∏∏ ν     i =1 {q} q ,i 

1

n Nq

= 1

This high-temperature product rule imposes the isotope fractionation to be nil at very hightemperatures. If we take advantage of this rule, Equation (9) then becomes:  3N exp ( − hν q ,i / 2kT ) 1 − exp ( − hν′q ,i / kT )  ν  = β AX ∏∏ q ,i × ×  i =1 {q} ν′q ,i 1 − exp ( − hν q ,i / kT ) exp ( − hν′q ,i / 2kT ) 

1

nN q

(10)

Liquid phases exhibit a higher degree of complexity, in particular because of the absence of long-range translational order and because of their dynamical behavior. More approximations are needed. The isotopic properties of liquid phases can be determined by adopting either the same approach as for crystals (i.e. building a periodic model), or the approach developed for gaseous molecules (i.e. building an isolated molecular cluster of variable size). The latter method can be justified for dissolved molecules that remain more or less intact in solution or for aqueous complexes where intra-complex bonds are probably much stronger than interactions with bulk solvent. The additional complexities that arise in dealing with liquid phases will be discussed in a later section.

Equilibrium Fractionation: A Molecular Modeling Perspective

33

Approximate formula based on force constants The above equations relate conveniently the isotopic fractionation factor to the vibrational frequencies but Bigeleisen and Mayer (1947) also derived a series of approximate formulae that are useful when all vibrational frequencies are not available and also for improving our understanding of the parameters that control equilibrium isotopic fractionation between two phases. Thus, if the frequency shift associated with the isotopic substitution is small and if the reduced energy is small (i.e. hn/kT ≲ 2) then Equations (7) or (10) become:

1 ∑ β AX =+ i

h 2 ∆ν i2 24(kT )2

Treating the vibrations as harmonic, squared vibrational frequencies can be related to the force constants and masses. By doing so, the reduced partition function ratio can be expressed as a function of the force constants acting on the element of interest: 2

F  m − m '  h  β AX =+ 1   2 π  24(kT )2 mm '   

(11)

F is the sum of force constants in three orthogonal directions opposing displacement of the atom X from its equilibrium position. If atoms X are located in more than one crystallographic site, the force constant for all sites must be averaged. The full derivation of the approximate formula (11) from Equations (7) or (10) can be found for instance in Young et al. (2015). This expression is a valid approximation for Equation (10): (i) at relatively high temperature (hν/kT < 2 implies, for instance, a temperature higher than 360 K if the vibrations involving atom X in phase A have wavenumbers, ω=ν/c, smaller than 500 cm−1 (1 cm–1 is equivalent to 30.0 GHz), or a temperature higher than 720 K if the relevant vibrations extend up to 1000 cm−1), (ii) when the difference in mass between the two isotopes is sufficiently small relative to the average atomic mass (this assumption excludes the very light elements such as hydrogen), (iii) assuming isotope-independent force constants; it is also worth noting that Equation (11) still assumes a harmonic vibrational partition function. The expression (11) clearly shows that, under the conditions of validity just mentioned, equilibrium isotopic fractionation varies proportionally to the reciprocal of the square of the temperature. When the reduced energy hν/kT becomes much higher than 2 (i.e. at low temperature or for high-frequency vibrations), it can be shown that the temperature dependence of fractionation factor weakens and tends to a 1/T behavior. This leads to a concave-down curvature of β-factors when plotted against 1/T2 and this curvature is more pronounced as one moves away from the conditions of validity of Equation (11). The mass-dependence of the same equation indicates that isotopic fractionations become smaller for heavy elements. Equation (11) also predicts that, at equilibrium, the heavy isotopes of an element will concentrate in the phase where the force constants are the greatest, i.e. in the phase where the element of interest involves the stiffest bonds (e.g., element in higher oxidation state, with lower coordination number). The two first points, i.e. temperature dependence and mass dependence, are well illustrated by the iron and oxygen β-factors of the goethite (Fig. 1). In this iron oxyhydroxide, vibrations involving iron atoms correspond to wavenumbers smaller than 600 cm−1, which dictates that Equation (11) is valid above ~ 400 K. Even at lower temperatures (i.e. in the stability field of goethite), Figure 1 shows that the departure of this approximate formula from Equation (10) is smaller than 0.5‰. Because oxygen is much lighter that iron, the equilibrium isotope fractionation factors are much larger (i.e. oxygen β-factor is ~6 times larger than iron β-factor). In goethite, half oxygen atoms are hydroxylated. The bending and stretching vibrational modes of these OH groups (observed at ~ 800 cm−1 and above 3000 cm−1, respectively) contribute

Blanchard, Balan & Schauble

34

T (K)

T (K) 773

473

373

323

298

773

273

12

473

373

323

298

273

OH

120

10 lnβ ( O/ O)

OH

18

16

80

O

60

O

3

6

3

57

54

10 ln! ( Fe/ Fe)

100 9

40

3

20 0

0 0

3

6 6

9 2

-2

10 /T (K )

12

0

3

6 6

9 2

12

-2

10 /T (K )

Figure 1. Temperature dependence of the iron (left) and oxygen (right) β-factors of goethite (α-FeOOH). Results obtained using the approximate formula (11) based on force constants (dashed lines, unpublished data) are compared with the results given by Equation (10) using all vibrational frequencies (solid lines, Blanchard et al. 2015).

to the oxygen β-factor. This explains why, in the case of the hydroxylated oxygen atoms, Equation (11) is not a valid approximation and meets the “correct” β-factor curve only at very high temperature whereas the approximation still holds for the other oxygen atoms. In first-principles calculations based on quantum-mechanics, the determination of β-factors using Equations (7) or (10) requires computing all vibrational frequencies, which is computationally expensive, whereas only electronic energy calculations performed for a limited number of positions of the atom of interest in the vicinity of its equilibrium position are needed to apply the approximate formula (11). When the conditions of validity stated above are met, the use of Equation (11) allows consideration of more phases, and phases with greater structural complexity. This includes liquids and crystal defects, which may require large model systems (i.e. more than a hundred of atoms). This approach has been applied for discussing the Cr isotope fractionation in conditions relevant to the differentiation of the Earth’s core (Moynier et al. 2011), the Li isotope fractionation between minerals and aqueous solutions at high pressure and temperature (Kowalski and Jahn 2011), and the S isotope fractionation of sulfate groups incorporated in major calcium carbonates (Balan et al. 2014). Without performing first-principles calculations, a qualitative estimation of the fractionation factors between phases where the element of interest involves contrasted bonding schemes, can be obtained by determining the force constants from an ionic model. In this model based on Pauling’s rules, the force constant is assessed mainly from the valences and ionic radii of the central element and its first neighbors. A description and application of the method for nontraditional isotopes is presented in Young et al. (2015). This ionic model derives from earlier studies that demonstrated and used the correlation existing between bond types and oxygen isotope fractionation in silicate minerals (e.g., Taylor and Epstein 1962; Garlick 1966; Schütze 1980; Richter and Hoernes 1988; Smyth and Clayton 1988). Even if this type of ionic model does not show a great accuracy, it highlights the basic crystal chemical parameters that govern the equilibrium stable isotope fractionation, by affecting the stiffness of interatomic bonds.

Equilibrium Fractionation: A Molecular Modeling Perspective

35

MODELING APPROACHES Quantum-mechanical molecular modeling As shown in the previous section, the determination of the equilibrium isotope fractionation factors can be related to the change of vibrational frequencies associated to the isotopic substitution. These vibrational frequencies can be calculated from empirical force fields built using experimental measurements like structural parameters, elastic properties or known vibrational frequencies. A review of these methods can be found in Schauble (2004). The alternative approach that spread out during the last decade thanks to the advances in processor speed and memory size, consists in using quantum-mechanical molecular modeling. We will give here only a short introduction aiming at helping experimental geochemists to approach these theoretical tools but many general or specialized publications are available elsewhere, like for instance in previous Reviews in Mineralogy and Geochemistry volumes (Cygan and Kubicki 2001; Perdew and Ruzsinszky 2010). The properties of any material can in principle be obtained from the laws of Quantum Mechanics by solving the equations describing the interactions between nuclei and electrons. In practice, a number of approximations are needed to address this complex problem. The first is the Born–Oppenheimer approximation, which considers that the rapid motion of electrons is decoupled from the slower motion of nuclei. Electronic wavefunctions are obtained by solving the Schrödinger equation for a system to which the positions of the nuclei are fixed external parameters. The energy of the system is then a function of the nuclei positions and the nuclei dynamics can be described by considering their motions on a potential energy surface. In comparison, empirical force fields use analytic functions to approximately describe the potential energy surface in terms of interatomic distance, oxidation state, effective ionic charge, and similar parameters that can be fitted by detailed examination of either experimental data or theoretical calculations. This parameterization of the potential energy surface may be assumed to be transferable, meaning that interatomic interactions in a group of related structures can be calculated using parameters fit to data from only one, or a subset of them. The reliability of empirical force fields will then be highly dependent on the quality of data available for parameter fitting, on the correct choice of structural variables to fit, the correct choice of suitable analytic functional forms, and on the consistency of electronic structure in the group of substances to which the force field is applied. Because of this chain of assumptions, and the use of empirical data, it can be difficult to assess the suitability of an empirical force field for calculating isotope fractionation factors. In addition, spectroscopic-quality force fields are not always available for substances of interest, especially for compounds and molecules containing heavy elements, unusual structures, or less common oxidation states. For these reasons, the information obtained using first-principles calculations is often more straightforward to generate, and easier to test against known vibrational and structural properties, than the outputs of analytic potentials. Against this caution, however, it should be noted that typical force-field parameterizations are much more mathematically efficient than electronic structure calculations, making it possible to probe systems with large numbers of atoms and/or dynamical disorder (such as liquids or traceelement substituted crystals) with relatively modest computational effort. In the quantum mechanical treatment, the Schrödinger equation of a multiple-electron system is most often solved using one of two different schemes. The Hartree–Fock (HF) method (Roothaan 1951) aims at determining the best multi-electronic wavefunction by combining mono-electronic wavefunctions (the so-called orbitals). The multi-electronic wavefunction exactly obeys the Pauli exclusion principle, whereas Coulombic interactions between different electrons are treated in a mean field approximation. It can be shown that the exact system energy is always lower than the Hartree–Fock energy, the difference being often referred to as the correlation energy. Instead of focusing on wavefunctions, density functional theory (DFT)

36

Blanchard, Balan & Schauble

(Hohenberg and Kohn 1964) is based a theorem requiring that all the ground state properties of a system of electrons moving under the influence of an external potential are uniquely determined by its electron density. Therefore, the ground state energy is a functional of the electronic density. Hohenberg and Kohn (1964) also demonstrated that the ground state energy can be obtained variationally because only the exact ground-state density minimizes this functional. Kohn and Sham (1965) proposed a practical scheme to build this functional by showing that a system of N interacting electrons can be treated as a fictitious system of N electrons that do not interact with each other but operate in an effective external potential taking into account an exchangecorrelation term. The corresponding mono-electronic equations, called Kohn–Sham equations, can be solved via an iterative and self-consistent procedure starting from an arbitrary electron density. This procedure should lead to the density that minimizes the energy (i.e. the exact ground state electronic density). Unfortunately, the exact expression of the exchange-correlation potential is unknown and approximate expressions have to be used. Two commonly used approximations are the local density approximation (LDA, based for example on a homogeneous electron gas) and the generalized gradient approximation (GGA) taking partial account of non-homogeneous effects. DFT is popular in part because it provides a description of the electronic ground state of many systems that is more accurate than standard HF methods, at a similar computational cost (for molecules). In this way, DFT makes it possible to efficiently model the static or dynamic properties of relatively complex systems, such as periodic systems containing up to few hundred atoms per unit cell. Unlike HF-based methods, however, there is not (or at least not yet) a welldefined hierarchy of post-GGA theories that can be used to systematically improve the accuracy of DFT models. Partial corrections for some known shortcomings in standard DFT functionals are well established and effective, such as the “DFT + U” technique for improving DFT models of transition-element oxides (Anisimov et al. 1991; Cococcioni and de Gironcoli 2005) or the various methods for including the dispersion interactions into DFT (e.g. Grimme 2011), however. In practice, electronic wavefunctions are represented using a finite set of fixed functions. These functions can be localized on the atomic positions (as are the atomic orbitals) or can consist of plane waves (which correspond to solutions of the Schrödinger equation for a free particle). Although not a stringent rule, localized basis sets are well suited for isolated molecules or clusters of molecules; whereas plane-waves are more appropriate to treat extended and periodic systems, such as crystalline solids. Localized basis sets make it possible to use hybrid DFT–HF methods such as the Becke three parameter Lee–Yang–Parr (B3LYP) method (Lee et al. 1988; Becke 1993), designed to emphasize the best features of each both theories. Hybrid methods are most commonly used to model the structure and vibrational frequencies of molecules. In order to reduce the computation cost without losing accuracy, it is also possible to restrict the explicit electronic structure calculations to the valence electrons because chemical properties mostly involve changes in the distribution of valence electrons. In this simplified treatment, the potential created by the atomic nucleus and core electrons is replaced by a pseudopotential. Pseudopotentials are most commonly used in conjunction with plane-wave basis sets for elements with Z > 2, or in localized basis function calculations involving elements with Z ≥ 20. Many different types of pseudopotentials have been developed, and high-quality public libraries of basis sets and pseudopotentials for almost all naturally occurring elements are now available online (e.g., GBRV, http://www.physics.rutgers.edu/gbrv/, Garrity et al. 2014; SSSP, http://materialscloud.org/sssp/; EMSL Basis Set Exchange, bse.pnl.gov/bse/portal, Schuchardt et al. 2007). These theoretical methods are implemented in numerous commercial and open-source software packages such as ABINIT (Gonze et al. 2002), CASTEP (Clark et al. 2005), CRYSTAL (Dovesi et al. 2014), GAMESS (Schmidt et al. 1993), Gaussian (Frisch et al. 2009), NWChem (Valiev et al. 2010), Quantum ESPRESSO (Giannozzi et al. 2009), or VASP (Kresse and Furthmüller 1996).

Equilibrium Fractionation: A Molecular Modeling Perspective

37

There are typically three steps in first-principles calculations for obtaining the vibrational frequencies needed for the determination of isotope fractionation factors. In the first step, the minimum-energy static structure is determined via geometric relaxation. From an initial guess geometry, often the experimentally determined structure, the forces on each atom and the stress over the cell are calculated, and a refined guess structure is determined. This procedure continues iteratively until the residual forces and stress are sufficiently small. Once the minimum-energy configuration has been calculated, the second step is the determination of force constants for displacements of the atomic nuclei from their equilibrium positions. Finally, vibrational frequencies are determined by a calculation with model force constants and appropriate isotopic masses (Baroni et al. 2001). Isotope substitution is expected to have a negligible effect on electronic structure, so a matrix of force constants for the common isotope in a molecule or a crystal can be recycled to estimate vibrational frequencies of uncommon isotope-substituted species. This means that frequencies corresponding to isotopically substituted species can be calculated very rapidly (i.e., in a few seconds on a personal computer), even for very complex substances, once the force constant matrix has been determined. In first-principles methods, uncertainty in calculated frequencies is typically the main factor limiting the accuracy of calculated fractionation factors. As mentioned above, isotope effects on vibrational frequencies can be calculated self-consistently, using a single set of force constants for each system. The errors on the vibrational frequencies are expected to be highly systematic and largely cancel when calculating isotope frequency shifts. Méheut et al. (2009) showed that a systematic correction of n% on the frequencies induces a relative systematic correction on the logarithmic β-factors (ln β) varying between n% (at low temperatures) to 2n% (at high temperatures). The commonly used generalized gradient approximation is for instance associated with a systematic underestimation of ~ 5% of the harmonic vibrational frequencies. This would lead to a relative uncertainty of ~ 0.5‰ on a β-factor of 10‰. Two approaches are sometimes adopted for correcting this systematic frequency error. In some studies, calculated frequencies are re-scaled to experimental ones in order to improve the accuracy of the calculated fractionation factors (e.g., Schauble et al. 2006; Black et al. 2007; Blanchard et al. 2009; Li et al. 2009; Méheut et al. 2009). We must however keep in mind that calculated frequencies are harmonic, as they should be when using equations based on the harmonic approximation (e.g., Eqns. (5) to (10)), while experimental frequencies are influenced by anharmonicity (Liu et al. 2010). In addition, the value of the scaling parameters may be associated with significant uncertainty, depending on the quality and precision of spectroscopic data available for the compound of interest. Some studies that focus on crystals choose to correct the theoretical results by fixing the unit cell parameters to the experimental values and by optimizing only the atomic positions (e.g., Kowalski and Jahn 2011; Blanchard et al. 2015; Pinilla et al. 2015), but this procedure will usually not completely correct systematic errors in the electronic structure method, and it will of course not be applicable in materials where unit cell parameters are not known a priori.

Theoretical studies of non-traditional stable isotope fractionation A big advantage to quantum-mechanical molecular modeling is the ability to derive a wide range of electronic, structural, energetic, vibrational properties from the same model. These properties can often be directly compared with observations to test the accuracy of the model. First-principles calculations also represent efficient tools to tackle crystal chemical parameters and mechanisms controlling isotopic fractionations. Over the past decade or so DFT studies have been applied to theoretical studies of stable isotope fractionation spanning most of the non-traditional stable isotopes systems represented in this volume. The results of these theoretical studies might best be discussed within the perspective of each system, considering isotopic measurements on natural and synthetic samples as well, and this will be done in the following chapters. Here we present a brief annotated bibliography in order of increasing atomic mass, highlighting some of these works:

38

Blanchard, Balan & Schauble • Lithium. Theoretical studies focused on the equilibrium fractionation of lithium isotopes in aqueous solution (Yamaji et al. 2001) and between aqueous fluids and various Li-bearing minerals such as staurolite, spodumene and mica (Jahn and Wunder 2009; Kowalski and Jahn 2011). Isotopic results were discussed in light of the speciation change of the aqueous lithium at high temperature and pressure. • Boron. Most of the first-principles studies investigated the equilibrium distribution of 10 B and 11B isotopes between boric acid and borate in aqueous solution at ambient conditions, motivated by the application of boron isotope composition of marine carbonates as paleo-pH proxy (Liu and Tossel 2005; Zeebe 2005; Rustad and Bylaska 2007; Rustad et al. 2010b). Tossel (2006) studied the isotopic fractionation associated with the boric acid adsorption on humic acids, and more recently Kowalski et al. (2013) investigated the B isotope fractionation between minerals, such as tourmaline and micas, and boron aqueous species at high pressure and temperature. • Magnesium. Black et al. (2007) studied the equilibrium Mg isotope fractionation in chlorophylls. This and several later studies made efforts to improve methods to determine isotopic fractionation in liquids, with a particular focus on the fractionation between aqueous Mg2+ and Mg-bearing carbonate minerals (Rustad et al. 2010a; Schauble 2011; Pinilla et al. 2015, Schott et al. 2016). Mg isotopes in mantle silicates were treated in Schauble (2011), Huang et al. (2013) and Wu et al. (2015b). • Silicon. Méheut et al. (2007, 2009, 2014) computed the equilibrium Si isotope fractionation factors in various silicate minerals, including phyllosilicates. Their data analysis enabled to identify the key structural and chemical parameters controlling the isotopic signatures. Huang et al. (2014) and Wu et al. (2015b) applied the DFT method to silicate minerals of the Earth’s mantle. Some DFT calculations coupled with isotopic measurements on meteorite and terrestrial samples focused on Si isotope fractionation between metal and silicates, in order to discuss the composition of the Earth’s core and the Earth formation (e.g., Georg et al. 2007; Ziegler et al. 2010). The equilibrium fractionation in silicic acid and its potential application as proxies for paleo-pH were investigated in Dupuis et al. (2015) and Fujii et al. (2015). He and Liu (2015), He et al. (2016) complemented equilibrium Si isotope fractionation factors among minerals, organic molecules and the H4SiO4 solution. Javoy et al. (2012) determined the Si isotope properties of small gaseous molecules and crystalline compounds in the cosmochemical context of the solar nebula. • Calcium. Theoretical Ca isotope fractionation factors between minerals and solution are presented in Rustad et al. (2010a), in Colla et al. (2013), and among pyroxenes in Feng et al. (2014). Griffith et al. (2008) estimated fractionation factors between barite and calcite. • Vanadium. Wu et al. (2015a) explored how V isotope fractionation depends on crystalchemical parameters such as valence, bond length and coordination number. They considered several inorganic V aqueous species and the adsorption of V5+ to goethite, by adopting a cluster model with explicit solvation shells. • Chromium. Schauble et al. (2004), and Ottonello and Zuccolini (2005) computed the equilibrium Cr isotope fractionation factors of some molecules in the system Cr–H–O–Cl as well as in magnesiochromite (Ottonello et al. 2007). Moynier et al. (2011) extended these theoretical predictions to additional Cr-bearing minerals. These latter data associated with isotopic measurements on a range of meteorites suggest that Cr depletion in the bulk silicate Earth relative to chondrites results from its partitioning into Earth’s core.

Equilibrium Fractionation: A Molecular Modeling Perspective

39

• Iron. Several DFT studies focused on isotopic fractionation among Fe species in aqueous solution (Anbar et al. 2005; Domagal-Goldman and Kubicki 2008; Hill and Schauble 2008; Ottonello and Zuccolini 2008, 2009; Hill et al. 2010; Fujii et al. 2014), others looked at iron-bearing minerals such as hematite, goethite, pyrite and siderite (Blanchard et al. 2009, 2010, 2015). Rustad and Yin (2009) investigated the isotopic properties of ferropericlase and ferroperovskite in lower-mantle conditions to discuss the Earth accretion and differentiation. Fe isotope fractionation between mineral and aqueous solution was the object of the studies by Rustad and collaborators (Rustad and Dixon 2009; Rustad et al. 2010a). Moynier et al. (2013) estimated the magnitude of the isotopic fractionation between different Fe species relevant to the transport and storage of Fe in higher plants. In addition to all these first-principles calculations, Mössbauer spectroscopy (e.g., Polyakov 1997; Polyakov and Mineev 2000) and nuclear resonant inelastic X-ray scattering, NRIXS (e.g., Polyakov et al. 2005; Dauphas et al. 2012) represent alternative techniques for obtaining Fe reduced partition function ratios. • Copper. Seo et al. (2007) determined the equilibrium isotope fractionation of Cu+ complexes relevant of hydrothermal ore-forming fluids. Sherman (2013) modeled Cu-bearing minerals and various aqueous Cu+ and Cu2+ complexes to predict the equilibrium isotopic fractionation of Cu resulting from oxidation of Cu+ to Cu2+ and by complexation of dissolved Cu. Additional Cu complexes were considered in Fujii et al. (2013a, 2014). • Zinc. Several theoretical works studied the isotope fractionation of Zn between various aqueous zinc complexes including aqueous sulfide, chloride, and carbonate species relevant to hydrothermal conditions (Fujii et al. 2010, 2011, 2014; Black et al. 2011). Other complexes were modeled to discuss the Zn isotope fractionation in roots and leaves of plants (Fujii and Albarède 2012). • Germanium. Li et al. (2009) determined equilibrium fractionation factors for a range of Ge-bearing compounds (aqueous species and minerals) simulated using cluster models. Li and Liu (2010) investigated the fractionation associated with Ge adsorption onto Fe(III)-oxyhydroxide surfaces. A cluster model was used to model the adsorption complex. Such adsorption processes occur in many environments, and thus may influence significantly the Ge isotope global budged. • Selenium. Equilibrium Se isotope fractionation factors of inorganic and organic Sebearing species in gaseous, aqueous and condensed phases were computed (Li and Liu 2011). • Strontium. A combined theoretical and experimental study focused on the Sr isotope fractionation during inorganic precipitation of barite, where several Sr-bearing minerals and crystalline strontium hydrates were modeled (Widanagamage et al. 2014). This work was preceded by the determination of the isotopic fractionation between SrO2 and a Sr2+ aqueous species (Fujii et al. 2008). • Molybdenum. Mo isotope fractionation factors were determined using a cluster approach, for many aqueous species including several forms of molybdic acid and polymolybdate complexes (Tossel 2005; Weeks et al. 2007, 2008; Wasylenki et al. 2008, 2011). These results confronted with experimental data aim at identifying the molecular mechanisms responsible to the Mo isotope fractionation during adsorption to manganese oxyhydroxides, which is a primary control on the global ocean Mo isotope budget. • Cadmium. Yang et al. (2015) computed using DFT the equilibrium isotopic fractionation factors for Cd species relevant to hydrothermal fluids.

40

Blanchard, Balan & Schauble • Rhenium. Theoretical Re isotope fractionation has recently been investigated by Miller et al. (2015). They especially assessed the magnitude of nuclear volume fractionation with respect to mass dependent fractionation. • Mercury and Thallium. Schauble (2007) performed first-principles calculations on these very heavy elements and could show that isotopic variation in nuclear volume is the dominant cause of equilibrium fractionation, exceeding mass-dependent fractionations. This is supported by two more recent works by Fujii et al. (2013b) and Yang and Liu (2015). Wiederhold et al. (2010) performed additional theoretical calculations, quantifying the relationship between ionic bonding and equilibrium mercury isotope fractionation. • Uranium. Abe and his collaborators (Abe et al. 2008a,b, 2010, 2014) investigated the uranium isotope fractionations caused by nuclear volume effects.

Modeling isotopic properties of liquid phases Many natural processes involve the participation of fluids. Isotopic signatures of minerals are very often related to fluid–rock interactions. Understanding the isotope fractionation processes between minerals and fluids is then of great importance. This understanding will include our ability to produce reliable theoretical mineral–solution isotopic fractionation factors. However calculations of fractionation properties of liquids and solvated elements under thermodynamic equilibrium represent a bigger challenge than for gaseous molecules or minerals. In an aqueous solution, an ion or molecule dissolved in water will interact with water molecules and other dissolved species in a continuously changing arrangement of hydrogen bonds and ion pairs. This disordered and dynamic character complicates significantly the problem. Determining the vibrational frequencies of such systems from first-principles calculations has a computation cost far greater than for minerals. Additional approximations are needed and can include the use of molecular clusters of finite size or the use of relaxed configurations from molecular dynamics simulations. Most of the theoretical predictions of isotope fractionation in aqueous species are based on the cluster approximation (e.g., Yamaji et al. 2001; Anbar et al. 2005; Black et al. 2007, 2011; Seo et al. 2007; Domagal-Goldman and Kubicki 2008; Hill and Schauble 2008; Ottonello and Zuccolini 2009; Fujii et al. 2010, 2014, 2015; Li and Liu 2010; Rustad et al. 2010a,b; Sherman 2013). In this case, the ion or molecular complex of interest is surrounded by water molecules forming a solvation shell and the whole is sometimes immersed in a continuum approximating the dielectric properties of the solvent. The stable structure of this isolated nanodroplet is obtained at T = 0 K by minimizing the forces acting on the atoms and the reduced partition function ratio (β-factor) is computed from the vibrational frequencies obtained in the harmonic approximation. The inclusion of the first solvation shell around the considered species is a first step towards the consideration of the solvation effect, i.e. effect explaining that most gases exhibit measurable isotopic fractionations between the vapor phase and solution. This approach is however hindered by several difficulties, such as the number of water molecules that must be included, the symmetry of the cluster, and the consistency between different aqueous species or between the aqueous species and the mineral. Let’s take the example of iron isotopes. First-principles calculations performed on small clusters give equilibrium isotopic fractionation between Fe(H2O)63+ and Fe(H2O)62+ of 2.5−3‰ at 22 °C for the isotopic ratio 56Fe/54Fe (Anbar et al. 2005; Domagal-Goldman and Kubicki 2008; Hill and Schauble 2008). These values are in good agreement with the experimental value of 3.00 ± 0.23‰ (Welch et al. 2003), even if the theoretical value depends on the cluster symmetry chosen. Here the two iron species only differ from the charge and are treated in a consistent way, which allows a cancellation of errors. However when the same theoretical data are combined with mineral β-factors (Polyakov and Mineev 2000; Polyakov et al. 2007; Blanchard et al.

Equilibrium Fractionation: A Molecular Modeling Perspective

41

T (K) 373

323

295

273

4

) + aq e2 (

)-F 3+ (aq Fe

2

56

54

10 ln! ( Fe/ Fe)

3

1

3

Fe3+(aq)-Hematite 0

Fe2+(aq)-Siderite -1 8

10 6

12 2

14

-2

10 /T (K )

Figure 2. Left: Example of molecular cluster used by Rustad et al. (2010a) to model aqueous Fe2+ and Fe3+. Right: Calculated (curves) and measured (circles) fractionations for the pairs Fe3+(aq)–Fe2+(aq), Fe3+(aq)–hematite and Fe2+(aq)–siderite. Theoretical β-factors are from Rustad et al. (2010a) for aqueous ions, from Polyakov and Mineev (2000) for siderite, and from Polyakov et al. (2007) for hematite. These two latter Mössbauer-derived β-factors are consistent with DFT results (Blanchard et al. 2009). Experimental data are from Skulan et al. (2002), Welch et al. (2003) and Wiesli et al. (2004).

2009), the calculated fractionations for Fe3+-hematite and Fe2+-siderite are in disagreement with experimental data. Preceded by few theoretical works emphasizing the importance of explicitly treating secondary solvation shells (e.g., Schauble et al. 2004; Liu and Tossell 2005), Rustad et al. (2010a) could show that the β-factors can be reliably computed from systems as small as M(H2O)62+ but when they are embedded in a set of fixed atoms representing at least the second solvation shell (Fig. 2). Furthermore their results suggest that the aqueous cluster is much more sensitive to improvements in the basis set than the calculations on the mineral systems. By applying these results, Rustad et al. (2010a) obtained more accurate β-factors for aqueous Fe2+ and Fe3+ and could reconcile theory and experiment for the mineral–solution fractionations (Fig. 2). Obviously, an observed disagreement between theory and experiment may have other reasons, such as kinetic effects during nucleation and crystal growth that could make the equilibrium assumption invalid. For example, if minerals form via oligomers or clusters as an intermediate step between the aqueous species and the mineral, then using a model of the bulk aqueous species will never reproduce the observed fractionation (e.g. Domagal-Goldman et al. 2009). More generally this cluster approach can be justified for dissolved molecules that remain more or less intact in solution (e.g., [ClO4]–, B(OH)3 and CCl4) or for aqueous complexes where intra-complex bonds are probably much stronger than interactions with bulk solvent (e.g., [Cr(H2O)6]3+, [FeCl4]–, and Mg2+ in chlorophyll). On the other hand, this method has the disadvantage of neglecting the constant exchange of particles within the solvation shells and other effects, such as the formation of chemical bonds and structural rearrangements as a function of temperature and pressure considered to be important for the calculation of the isotope fractionation. To account for these dynamical phenomena, i.e. frequent particle exchange in the hydration shell and structural evolution of the fluid with pressure and temperature like it is for instance the case in Li aqueous solution (Jahn and Wunder 2009) one can go beyond the static calculations on

42

Blanchard, Balan & Schauble

molecular clusters by employing molecular dynamic simulations. In this case, a first-principles molecular dynamics is run at finite temperature where the condensed phase of the fluid is described through periodic boundary conditions. The equilibrated trajectory thus provides a representative distribution of the configurational environments of the species of interest in the fluid. The average fractionation factor is then estimated from the harmonic vibrational frequencies computed on a set of uncorrelated snapshots taken from the molecular dynamic trajectory. Vibrational frequencies can be computed directly from each snapshot without relaxing the atomic positions but this raises a problem because the dynamic structures are often statically unstable, meaning that some calculated frequencies are imaginary numbers. It is not clear how to make reliable thermodynamic calculations of fractionation factors when imaginary frequencies are encountered. A way around this problem is to further process each snapshot structure by allowing atomic positions to relax into the nearest local energy minimum by performing geometry optimization at T = 0 K (giving the so-called inherent structures, Stillinger and Weber 1983) before computing vibrational frequencies. This approach is more satisfactory for determining the fractionation properties from the equations based on the harmonic approximation, but on the other hand this approach erases some of the desired dynamical sampling, evident for instance in the more homogeneous bond lengths found after snapshot relaxations. An alternative approach was proposed by Kowalski and Jahn (2011) and consists in relaxing only the position of the element of interest before determining the fractionation properties from the high-temperature approximation based on force constants. When this approximation is valid, it reduces significantly the computational cost, which is the major drawback of the molecular dynamics method. In order to keep the calculations as tractable as possible, all parameters must be chosen carefully, including the size of the simulation cell and the snapshot sampling. The simulation cell should be large enough to avoid significant interaction between atoms and their periodic images. The first studies of this kind used simulation cells containing typically 32 or 64 water molecules (Rustad and Bylaska 2007; Kowalski and Jahn 2011; Pinilla et al. 2014, 2015; Dupuis et al. 2015). However Kowalski and Jahn (2011) have shown that for a dissolved Li+ ion a cell containing only 8 water molecules is enough to get a converged result within the accuracy of the calculations. This highlights the local character of fractionation properties, i.e. isotopic fractionation is mainly controlled by the bonds formed with the first atomic neighbors. The snapshot sampling should be large enough to get a representative distribution of the fluid configurations but small enough to keep the computation time under reasonable limits. Dupuis et al. (2015) tested thoroughly this sampling by considering a random, periodic or selected extraction of the snapshots. Results suggested that the extraction of only 10 snapshots is statistically representative of the whole solution, and that this number can even be decreased by taking advantage of the correlation between the fractionation value and the mean bond length (in cases where such correlation is evidenced). The first study taking into account dynamical effects on isotope fractionation factors for non-traditional elements is by Rustad and Bylaska (2007). They calculated first the velocities correlation of exchanging isotopes and through its Fourier transform found the vibrational density of states to predict the boron isotope fractionation between B(OH)3 and B(OH)4− in aqueous solution. This led to a discrepancy between the calculated fractionation factor and the experimental one (Byrne et al. 2006; Klochko et al. 2006), which was solved after computing the harmonic frequencies of inherent structures taken from the molecular dynamics trajectory. Kowalski and coworkers took advantage of computing partial vibrational properties to investigate the lithium and boron isotope fractionation between aqueous fluids and minerals at high pressure and temperature (Kowalski and Jahn 2011; Kowalski et al. 2013). More recently, Pinilla et al. (2015) studied the equilibrium isotope fractionation between aqueous Mg2+ and carbonate minerals, and Dupuis et al. (2015) focused on silicon isotope fractionation in dissolved silicic acid. In conclusion, first-principles molecular dynamics simulations represent an efficient way to take into account the dynamical aspect of the fluid and their compressibility. By employing periodic boundary conditions, this approach also allows to treat minerals and

Equilibrium Fractionation: A Molecular Modeling Perspective

43

fluids in a consistent manner; a prerequisite for reliable isotope fractionation factors between mineral and solution. All methods mentioned so far are based on the harmonic approximation. For many substances, uncertainties associated with calculated vibrational frequencies are likely to be larger than anharmonic effects. In liquids, anharmonicity effects are expected to have stronger impacts on fractionation properties. Generally anharmonicity will tend to decrease the vibrational frequencies and consequently the reduced partition function ratios (e.g., Richet et al. 1977; Balan et al. 2007; Méheut et al. 2007). To go beyond the harmonic approximation, more sophisticated techniques exist and are presented in the next section.

Beyond harmonic approximation: Path integral molecular dynamics As already pointed out, isotopic fractionation is a quantum effect. Nuclear quantum effects (e.g., zero-point energy, quantum tunneling) whose relative contribution increases with decreasing temperature, influence significantly the properties of many systems, especially those containing lighter elements. Moreover it is also known that anharmonicity can be substantial especially for light elements and for liquid phases. A method of choice to include quantum nuclear effects without using the harmonic approximation is the method of thermodynamic integration coupled to path integral molecular dynamics (PIMD). The reduced partition function ratio can be written using the Helmholtz free energy instead of the partition function:

= ln β AX

F ( AX ) − F ( AX ')  F ( AX ) − F ( AX ')  +  kT kT  cl

(12)

where F(AX) and F(AX') are the free energy of a single molecule of the two isotopologues AX and AX', and the subscript cl refers as before to quantities calculated using classical mechanics. Unfortunately, the absolute value of the free energy is not a quantity that can be directly obtained for any arbitrary system. Relating the free energy to another physical property, such as the kinetic energy, can circumvent this problem. On this line, it can be shown that the free energy of an isotopic species depends on its kinetic energy and mass (Landau and Lifshitz 1980): ∂ (13) = − ∂m m where 〈〉 represents a thermodynamic average in the canonical ensemble (i.e. thermodynamic ensemble NVT corresponding to a system in thermal equilibrium: the number of particles, the volume and the temperature of the system are fixed). Inserting Eqn. (13) into Eqn. (12) and taking into account that in the classical limit the kinetic energy of an atom is 〈K〉 = 3kT/2, the β-factor is then given by: ln β= AX

1 kT

m

∫ dµ

m'

K µ µ

3 m − ln   2  m' 

(14)

where 〈K(µ)〉 is the average kinetic energy of the atom X of mass µ in phase AX. In this expression, the β-factor is thus obtained by thermodynamic integration from mass m' to mass m. Here, we stress that the kinetic energy used in the thermodynamic integration is that of the quantum system. It differs from the kinetic energy determined using standard molecular dynamic methods. These latter methods solve the classical equation of atomic motions in a force field, which can be defined either empirically or using ab initio electronic structure calculations. In the present case, the determination of the kinetic energy has to take into account the fact that, in a quantum system, the atomic trajectories are not defined. The atoms display some degree of delocalization (i.e. some uncertainty on their position); which is inversely related to their mass. Path integral methods enable the treatment of such effect by replacing the standard classical system by a larger number

Blanchard, Balan & Schauble

44

of replicated classical systems (Fig. 3). The replicated systems interact through harmonic springs connecting a given atom to its counterpart in adjacent replicas. Based on the exact isomorphism between a quantum particle and a classical ring polymer, the quantum thermodynamic averages can be calculated exactly for any force field using path integral methods. PIMD methods are implemented in several codes such as the freely available program CP2K, CPMD (Marx and Hutter 2000), i-PI (Ceriotti et al. 2014) or PINY_MD (Tuckerman et al. 2000). A description of the PIMD methods and their implementations is out of the scope of this chapter but can be found elsewhere (Feynman and Hibbs 1965; Ceperley 1995; Tuckerman 2010). The drawback of PIMD methods is the computational cost that is almost prohibitive for treating at the ab initio level most of the systems relevant in geosciences. To address this issue, several studies report new developments that improve the efficiency of the methods concerning isotopic applications (e.g., Ceriotti and Markland 2013; Cheng and Ceriotti 2014; Marsalek et al. 2014). Regarding the investigation of isotopic effects, many studies focused on small molecules or molecular clusters, including water molecule and ions, hydrated chloride ions, carbon dioxide, organic molecules (e.g., Tachikawa and Shiga 2005; Vanicek and Miller 2007; Suzuki et al. 2008; Mielke and Truhlar 2009; Pérez and von Lilienfeld 2011; Webb et al. 2014). Other studies have modeled condensed phases, like for instance Chialvo and Horita (2009), Ramírez and Herrero (2010), Markland and Berne (2012), Zeidler et al. (2012), and Pinilla et al. (2014) for the water system. Among these studies, Pinilla et al. (2014) determined the H and O isotope equilibrium fractionation between water ice, liquid and vapor, and compared the exact result obtained from PIMD with those of the more common modeling strategies, which involve the use of the harmonic approximation. The same approach was then applied to the aqueous Mg2+ (Pinilla et al. 2015). Results show the importance of including configurational disorder for the estimation of isotope fractionation in liquid phases, by using molecular dynamics simulations. In the case of D/H fractionation, neglecting the anharmonic effects leads to an overestimation of the fractionation factor. In other words, the harmonic approximation will overestimate the concentration of heavy isotopes in the aqueous phase. For heavier atoms, like magnesium and to some extent oxygen, methods based on the harmonic approximation give reliable results and in the same time reduce significantly the computational cost.

Standard Molecular Dynamics

Path Integral Molecular Dynamics

2 O

H

1

H

1

2 3

3 1

2 3

Figure 3. Schematic representations of a water molecule in standard molecular dynamics and path integral molecular dynamics. The straight lines joining the replicas (also called “beads”) with the same number represent the interatomic interactions that can be modeled using either empirical or ab initio force fields. Replicas belonging to the same atom interact through harmonic springs. Only three replicas are represented here for clarity reasons but a large number of replicas (several tens) are actually needed to capture the quantum behavior of the system.

Equilibrium Fractionation: A Molecular Modeling Perspective

45

MÖSSBAUER AND NRIXS SPECTROSCOPY In addition to the equilibrium fractionation factors derived experimentally by isotopic composition measurements, we have seen that these equilibrium constants can also be determined theoretically from the computation of the vibrational properties. An additional approach for Mössbauer-active elements (like iron, which is the most commonly studied element) consists in using Mössbauer spectroscopy (e.g., Polyakov 1997, 2000) and nuclear resonant inelastic X-ray scattering, NRIXS (e.g., Polyakov et al. 2005; Dauphas et al. 2012). These two latter techniques probe the vibrational properties of the target element and are thus ideally suited to study complex materials. Let’s start again from an expression relating the reduced partition function ratio (β-factor) to the kinetic energy. Using the first-order thermodynamic perturbation theory (Landau and Lifshitz 1980), Equation (14) becomes:

m − m'  K 3  (15) − m  RT 2  In Mössbauer spectroscopy, the kinetic energy K of the active isotope (e.g. 57Fe) is related to the second-order Doppler shift, S(T): = ln β AX

S (T ) = −

K (T )

mc where c is the light velocity. Substituting S(T) for K(T) into Equation (15) leads to: = ln β AX

m − m '  m 'cS (T ) 3  +  2 m  RT

The second-order Doppler shift S(T) can be determined experimentally from the temperature dependence of the isomer shift because both quantities only differ by a constant value that reflects the fact that the isomer shift is measured relative to a reference spectrum of metallic iron at room temperature. Experimental data are conveniently fitted using a Debye function: 4   T  θM T x 3 9Rθ M  1 + 8  S (T ) = − dx   ∫0 x e −1  16 mc   θM   

where m is the mass of the resonant isotope (57Fe), and θM is a characteristic Mössbauer temperature. However, the SOD shift is not the only factor controlling the temperature shift in the Mössbauer spectra, so model assumptions about the temperature dependence of the Mössbauer isomeric shift are needed. In practice, the prediction of Mössbauer-derived fractionation factors involves an extensive data processing and requires high quality data to achieve a reasonable accuracy. This explains the few cases of conflicting results and revised data reported in the literature (e.g., Polyakov et al. 2007; Rustad et al. 2010a; Blanchard et al. 2012). In the NRIXS method, kinetic energy is calculated from the measured vibrational density of states of the element of interest using the following expression. The vibrational density of states can also be obtained by ab initio calculations. 3 emax (16) E ( e / kT ) g(e)de 2 ∫0 where g(e) is the vibrational density of states of 57Fe, for instance, normalized to unity, and E(e/kT) is the Einstein function for the vibrational energy of a single harmonic oscillator at frequency ν = e/h. emax corresponds to the maximal energy of the vibrational spectrum, and the Einstein function is given by: K=

E ( e kT ) e kT e = + kT exp(e kT ) − 1 2kT

(17)

46

Blanchard, Balan & Schauble

Equations (16) and (17) are valid in the harmonic approximation and Equation (16) takes into account the virial harmonic relation K = Evib/2, where Evib is the vibrational energy of the harmonic oscillator. For a given system, the β-factor calculated from the vibrational density of states (Eqns. 15 and 16) is identical to the β-factor calculated directly from the vibrational frequencies (Eqn. 10). When the highest energy of the vibrational density of states is smaller than 2πkT, another expression can be used, based on a Bernoulli expansion of the β-factor (Dauphas et al. 2012): g  m4g m6g  m'  m ln β AX ≈  − 1   22 2 − + 4 4 6 6  m 8k T 480k T 20160k T   

(18)

where mig is the ith moment of the vibrational density of states g(e), given by: +∞

mig = ∫ g(e)ei de 0

Dauphas et al. (2012) have also shown that the even moments of g(e) can be obtained directly from the moments of the NRIXS spectrum S(e) and Equation (18) can be rewritten as:

( )

2 S S S S S S S  RS R S − 10 R2S R3S R7 + 210 R2 R3 − 35R3 R4 − 21R2 R5  m'  1 ln β AX ≈  − 1   23 2 − 5 + 480k 4T 4 20160k 6T 6 m  ER  8k T 

   

where ER is the free recoil energy and RiS is the ith moment of S(e) centered on ER, given by: = RiS

∫

+∞

−∞

S (e)(e − ER )i de

The NRIXS-based method probing directly the vibrational properties of the target element is expected to provide better accuracy for the β-factors than that based on Mössbauer spectroscopy. However NRIXS spectra have only recently been measured specifically for applications to isotope geochemistry (Dauphas et al. 2012). A difficulty that had been unappreciated before, was encountered concerning the baseline at low and high energies (the details of the spectrum at the low- and high-energy ends heavily influence the treatment of experimental data and therefore the value of the β-factor). To address this issue, Dauphas and collaborators have developed a software (SciPhon) that reliably corrects for non-constant baseline (Dauphas et al. 2014; Blanchard et al. 2015; Roskosz et al. 2015). Figure 4 displays a comparison of the iron β-factors derived from NRIXS, Mössbauer measurements and first-principles calculations (DFT) for various iron-bearing minerals, i.e. sulfides, an oxide and a carbonate. For siderite (FeCO3), DFT results (Blanchard et al. 2009) are in excellent agreement with Mössbauer data (Polyakov et al. 2007). The same kind of agreement is found between the DFT and NRIXS resuls of chalcopyrite (CuFeS2, Polyakov et al. 2013). For hematite (Fe2O3) DFT-derived iron β-factors (Blanchard et al. 2009) are very close to NRIXS-derived values (Dauphas et al. 2012) while the results from Polyakov et al. (2007) are slightly above. In the case of pyrite (FeS2), the apparent discrepancy between DFT and Mössbauer results that was reported in Blanchard et al. (2009) and in Polyakov and Soultanov (2011), could be resolved by using a better constrained temperature dependence of the Mössbauer spectra (Blanchard et al. 2012). The value of the iron β-factor in pyrite was confirmed by NRIXS data (Polyakov et al. 2013) and also appears consistent with experimental measurements of equilibrium isotopic fractionation between pyrite and dissolved Fe2+ (Syverson et al. 2013). These comparisons exhibit that DFT, NRIXS and Mössbauer spectroscopy should lead to statistically indistinguishable β-factors, when high-quality measurements are performed followed by a careful data processing. Therefore, the comparison of results from these independent techniques provides reliable isotope fractionation factors. Figure 4 also shows that

Equilibrium Fractionation: A Molecular Modeling Perspective

47

T (K) 773

373

323

298

273

DFT NRIXS Mössbauer

15

12

3

57

54

10 ln
( Fe/ Fe)

473

9

6

3

0

0

3

6 6

9 2

12

-2

10 /T (K )

Figure 4. Comparison of iron β-factors derived from first-principles calculations (DFT), NRIXS and Mössbauer measurements for siderite (FeCO3), chalcopyrite (CuFeS2), hematite (Fe2O3) and pyrite (FeS2). Data were taken from Blanchard et al. (2009, 2012), Dauphas et al. (2012), Polyakov and Mineev (2000), Polyakov and Soultanov (2011), Polyakov et al. (2007). In these minerals where iron atoms are always in octahedral coordination, the β-factor is mainly controlled by the iron oxidation state and the degree of covalence of the chemical bonds involved.

iron β-factors of these minerals are noticeably different; pyrite displaying the highest value while siderite has the lowest one. In all minerals except chalcopyrite iron atoms are in octahedral sites, the observed order can be discussed in terms of oxidation state (Fe3+ in hematite vs. Fe2+ in siderite) and degree of covalence of the interatomic bonds (low-spin, strongly covalent d-orbitals in pyrite vs. high-spin, ion-like d-orbitals in almost all other minerals).

MASS-INDEPENDENT FRACTIONATION AND VARIATIONS IN MASS LAWS Typical fractionating processes, including equilibrium isotope partitioning, activationenergy and transport-controlled disequilibrium reactions, and even gravitational and centrifugal isotope separation, almost always impart a characteristic signature in which the magnitude of isotope fractionation scales in close proportion to the difference in isotopic mass (e.g., Hulston and Thode 1965; Matsuhisa et al. 1978; Weston 1999; Young et al. 2002). A typical example is oxygen, where 17O/16O fractionation is usually 0.5−0.53 times as large as 18 O/16O fractionation, very close to the ratio of mass differences (≈0.501). In high temperature igneous and metamorphic rocks the mass-fractionation relationship for oxygen is remarkably consistent, with d17O ≈ 0.528 ± 0.001 × d18O (e.g., Rumble et al. 2007). Subtle variations in mass dependence are observed in light stable isotope systems, including oxygen and sulfur (Farquhar et al. 2003; Barkan and Luz 2012; Hofmann et al. 2012), and are of increasing interest as potential tools to unravel the nature of fractionation in the hydrological cycle, in the precipitation of low-temperature minerals from solution, and in biochemical reactions. Young et al. (2002) pointed out that variations in mass-fractionation relationships could also help distinguish equilibrium from kinetic fractionations in non-traditional elements such as magnesium, although such measurements are likely to require very high precision in systems where fractionations of only a few per mil are observed.

48

Blanchard, Balan & Schauble

Various notations have been developed to describe variations in mass dependence. Here we have followed the basic formulation of Mook (2000), which has been widely adopted. In a stable isotope fractionation involving element X with stable isotopes, 1X, 2X, and 3X that have masses m1, m2, and m3, there are two distinct fractionation factors α: 3/1

( (

) )

( (

) )

 3 X  /  1 X  α AX / BX =  3 X  /  1 X 

AX BX

and 2 /1

 2 X  /  1 X  α AX / BX =  2 X  /  1 X 

(19)

AX BX

In such a system it is convenient to express the mass dependence of the fractionation as a mass-fractionation exponent, θ, the ratio of the natural logarithms of the fractionation factors: θ=

( ln (

ln

2 /1

α AX / BX

3/1

α AX / BX

) )

(20)

Formally, mass-independent fractionation refers to any deviation of an observed fractionation from a reference mass fractionation exponent, typically expressed in delta notation: ∆ '2 1 X AX BX (‰ = ) 103 ( θ − θreference ) ln

(

31

α AX BX

)

(21)

Note that the prime indicates that logarithmic delta is being used here. Alternative expressions using conventional delta units may also be used, but for a discussion of fractionation factors the logarithmic delta makes the math simpler. The reference exponent might be a theoretical law or an empirically observed trend (an empirical exponent may be indicated by using λ instead of θ). There is generally not a well-accepted consensus in favor of a particular mass law exponent, so it can be tricky to compare ∆' values reported by different labs.

Variability in mass laws for common fractionations Detailed derivations of mass law exponents for various fundamental fractionation processes have been published elsewhere (e.g., Young et al. 2002; Dauphas and Schauble 2016), and will not be reproduced here. Several of the most significant and/or common mass law exponents are listed in Table 1, below, along with calculated exponents for some traditional and nontraditional stable isotope systems. Among the most important of these is the high-temperature equilibrium mass law exponent, which can be derived from the simplified formula for isotope fractionation in Equation (11), above (Matsuhisa et al. 1978):

θEq.,Hi-T

1 1 − m1 m2 ≈ 1 1 − m1 m3

(22)

As noted above for Equation (11), the constant mass dependence implied by this relationship is a surprisingly good approximation at low temperatures, even for materials with highfrequency vibrations such that hν/kT > 2 (Matsuhisa et al. 1978). In part, this occurs because the low-temperature equilibrium exponent is identical in the limit where the element of interest is much more massive than other atoms in the molecule (Cao and Liu 2011; Dauphas and Schauble 2016). These light-atom molecules tend to have the highest vibrational frequencies

Equilibrium Fractionation: A Molecular Modeling Perspective

49

Table 1. Theoretical mass-fractionation exponents. Type of fractionation

θ exponent

16,17,18

O

24,25,26

Mg

28,29,30

32,33,34

Si

S

54,56,57

Fe

198,200,202

Hg

Equilibrium, high-T

1 1 − m1 m2 1 1 − m1 m3

0.5305

0.5210

0.5178

0.5159

0.6780

0.5049

Equilibrium, low-T, light partner

1 1 − m1 m2 1 1 − m1 m3

0.5305

0.5210

0.5178

0.5159

0.6780

0.5049

0.5232

0.5160

0.5135

0.5121

0.6750

0.5037

Equilibrium, low-T, heavy partner†

1 1 − m1 m2 1 1 − m1 m3

Graham’s law (pinhole) effusion, atomic

m  ln  1   m2  m  ln  1   m3 

0.5158

0.5110

0.5092

0.5083

0.6720

0.5024

Graham’s law effusion, high-mass molecule

M  ln  1   M2  M  ln  1   M3 

0.5010

0.5010

0.5006

0.5007

0.6660

0.4999

Kinetic, transition state theory, jump limited

 µ*  ln  *1   µ2   µ*  ln  *1   µ3 

Gravitational/ centrifugal

m2 − m1 m3 − m1

Calculated variability

Intermediate between high-T equation and high-mass molecular diffusion

0.5010

0.5010

0.5006

0.5007

0.6660

0.4999

0.0295

0.0200

0.0172

0.0151

0.0119

0.0050

Note: mi are isotopic masses, Mi are masses of isotopically substituted molecules, and µi* are reduced masses of a reaction coordinate at the transition state. Based on Matsuhisa et al. (1978), Young et al. (2002), and Dauphas and Schauble (2016). † This equation only applies to the reduced partition function ratio βAX, and thus to fractionation relative to atomic vapor. Actual βAX exponents for non-traditional elements will rarely, if ever approach this limit because hν/kT and/or the masses of bond partners are too small.

50

Blanchard, Balan & Schauble

and hν/kT, especially when considering non-traditional (typically high atomic mass) isotope systems. This mass law is thus a common and sensible choice as a theory-based reference exponent (e.g., Young et al. 2002). The mass-dependent relationships described above indicate that there will be a narrow range of variability in mass dependence for typical fractionating processes. For equilibrium fractionations of non-traditional isotopes, it is expected that the variability will be quite small. Indeed, commonly occurring sulfur species (Z = 16) show very little change in mass dependence at equilibrium at relevant temperatures (Hulston and Thode 1965; Farquhar et al. 2003; Otake et al. 2008). There has not been a focused theoretical effort to quantify the variability in elements heavier than sulfur. However, studies of oxygen and sulfur suggest that even fairly crude electronic structure models (including DFT) can give an accurate picture of mass dependence variations at equilibrium (Cao and Liu 2011), and this seems likely to be an area of future development, as measurement precision continues to improve for many non-traditional elements.

Mass-independent fractionation in light elements (O and S) In addition to the subtle variations in mass dependence discussed above, there are some natural and laboratory environments that give rise to fractionations that deviate strongly from a proportional relationship with mass differences. These are called mass-independent fractionations, even though they are usually driven, ultimately, by differences in isotopic mass. The best-known examples of mass-independent fractionation are in oxygen and sulfur isotopes, and are thought to be associated with reactions between molecules in the gas phase. Large, approximately 1:1 variation in 17O/16O vs. 18O/16O is observed in primitive meteoritic oxides and silicates (Clayton et al. 1973). Although the cause of this fractionation is not yet settled, the most common explanation is that it represents a self-shielding effect in carbon monoxide, in which the common isotopologue 12 16 C O is optically thick to incoming light with the right energy to break it apart, while 12C17O and 12C18O are optically thin, and thus more prone to react in the interior of the solar nebula or a molecular cloud (Clayton 2002; Lyons and Young 2005). In the stratosphere, a large, ~1:1 fractionation of 17O/16O and 18O/16O is found in ozone, and in gases that exchange oxygen with ozone. It is thought that this fractionation reflects an isotopic effect on the lifetime of excited ozone molecules (Heidenreich and Thiemens 1986; Mauersberger 1987; Gao and Marcus 2001). Mass-independent sulfur isotope fractionation has been found widely in Archean and earliest Proterozoic samples (Farquhar et al. 2000). These samples show a range of 33S/32S vs. 34S/32S relationships, thought to be caused by photochemical reactions of SO2 in the early atmosphere, before O2 became a major constituent of air (Pavlov and Kasting 2002; Lyons 2007).

Mass-independent fractionation in non-traditional elements (Hg, Tl, and U) For non-traditional stable-isotope systems, at least two different fractionation mechanisms seem to be responsible for mass-independent fractionation effects. Most dramatic are large (> 1‰) mass-independent mercury isotope fractionations that are mainly photochemical (e.g., Bergquist and Blum 2007), and appear to be magnetic isotope effects dependent on the nonzero spins of the odd numbered mercury isotopes 199Hg and 201Hg. Introductory reviews of the magnetic isotope effect have been presented elsewhere (Turro 1983; Buchachenko 1995, 2013). The effect is apparent only in a subset of disequilibrium reactions. As yet, there are not any quantitative theoretical models that can reproduce the observed mass-independent signatures, and this is an area where more work is clearly needed. Another type of fractionation is observed in the uranium and thallium isotope systems (Stirling et al. 2007; Rehkämper et al. 2002). Variation in isotope abundances in these elements in nature appears to mainly result from an equilibrium mass-independent phenomenon: the nuclear field shift effect. This effect also acts to fractionate mercury isotopes (e.g., Schauble 2007; Estrade et al. 2009; Wiederhold et al. 2010; Ghosh et al. 2013), but the mass-independent signature is much more subtle than the largest MIFs observed in natural samples, e.g, Blum et al. (2014). This effect has been the subject of a number of first-principles theoretical studies.

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Figure 5. Nuclear field shift fractionations depend on the effect of the size and shape of a nucleus on the binding energy of electrons. The simplified example shown here assumes a single electron attracted to a spherical nucleus with a uniform charge density. The solid line shows the electrostatic potential binding an electron to an infinitesimally small nucleus, which goes to negative infinity as the electron approaches the nuclear center. For finite nuclei, the Coulomb potential does not go to negative infinity, but instead approaches a finite minimum inside the nucleus because the net electrostatic attraction to the shell of nuclear charge farther from the center than the electron is zero. The minimum is higher (the binding potential is weaker) for a large nucleus than for a small nucleus. Here the radius difference is assumed to be 10%, which is much larger than the difference between stable isotopes of most elements. Adapted from Schauble (2007).

Bigeleisen (1996) and Nomura et al. (1996) proposed that equilibrium isotopic fractionation in elements with very high atomic numbers could be driven by differences in the shape and size of nuclei, in addition to differences in mass. Hints of this effect were also described in an earlier experimental study of strontium isotope fractionation (Nishizawa et al. 1995). This is the nuclear field shift effect, and it is caused by overlap of electron density with the spatial volume occupied by the positive charge of a nucleus (Fig. 5). The general effect is to reduce the binding energy of electrons around large nuclei. This nuclear volume effect appears to be the most important component of the field shift, but non-spherical shapes may also be important for field shift effects in some nuclei; this shape-dependent part of the field shift is not as well studied as the volume component (e.g., Knyazev et al. 1999). Bigeleisen, Nomura, and their collaborators used the field shift effect to explain laboratory uranium isotope fractionation experiments in which the oxidized form of uranium, U(VI), had lower 238U/235U than coexisting reduced species at equilibrium. Such inverted redox/ fractionation relationships are rare. A key observation was that 236U/238U, 238U/234U, 238U/235U, and 238U/233U fractionations did not obey a consistent mass dependent relationship, with the magnitude of 238U/235U fractionation for instance being very similar to 238U/234U, despite a 3:4 mass difference ratio (Nomura et al. 1996). Isotope fractionations caused by the field shift effect can be quantified if field shift energies are known:

(

)

(

)

 E 0 ( AX ) − E 0 AX *  −  E 0 ( BX ) − E 0 BX *  (23)     ln α FS = kT where αFS is the field shift fractionation factor, and E0(AX), etc. are the ground state electronic energies of isotopic forms of AX and BX. Approximate expressions for the field shift energies have been derived in the optical spectroscopy literature (e.g., King 1984); these formulations

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capture the dependence of the field shift effect on the size of the nucleus (i.e., spatial nuclear volume) and on the electron density that overlaps it, e.g.,

(

)

2 πZe2 2 2 (24) Ψ (0) AX − Ψ (0) BX ∆ r 2 3kT where Z is the nuclear charge, e is the charge of an electron, |Ψ(0)AX|2 and |Ψ(0)BX|2 are the electron densities at the center of the nucleus of the atom of interest in substances AX and BX, and ∆〈r2〉 is the difference in mean-squared nuclear charge radius between the fractionating isotopes. This expression is approximate because it assumes a simplified model of the distribution of charge density in nuclei, that the nuclei are spherical (nuclei with odd numbers of neutrons and/or protons are aspherical) and that the electronic structure does not change as the nuclear size changes. The assumption of sphericity means that only the nuclear volume component of the field shift is considered, and potential shape effects are ignored. ln α FS ≈

Given these assumptions, however, it is clear that large nuclear field shift fractionations are most likely to occur between substances where the electron densities at the nucleus are very different, and where the difference in nuclear charge radius is large. Because the wavefunctions d- and f-orbital electrons (and most p-orbital electrons) do not overlap significantly with nuclei, variations in s-orbital electron population and structure control field shift fractionations. Based on Equation (24), and theoretical studies made so far, it is possible to list some qualitative rules of thumb about the chemical and physical properties that control field shift isotope fractionations: 1. Field shift isotope effects scale with nuclear charge radius, not with isotopic mass. Nuclear charge radii usually (but not always) increase with increasing neutron number, but tend to be smaller for nuclei with odd numbers of neutrons than one would expect from the radii of neighboring even-neutron number nuclei. For this reason, field shift fractionations will often generate a characteristic odd-even fractionation pattern (e.g., Bigeleisen 1996; Nomura et al. 1996). Not all elements show this pattern, however. For platinum, radii are almost perfectly linear for both odd and even numbered nuclei. In contrast, the 52Cr nucleus, with a “magic” number of 28 neutrons, is notably smaller than stable chromium isotopes with fewer or greater numbers of neutrons. 2. Changes in s-electron occupation and s-orbital shapes control field shift fractionation. 3. Species with more s-electrons, and more compact s-orbitals, will tend to attract smaller nuclei. Examples include Hg(0) vs. Hg (II) and Tl(I) vs. Tl(III)—in each case the reduced form has two 6s electrons while the oxidized form has none, so Hg(0) and Tl(I) species preferentially incorporate small (neutron poor) isotopes. This pattern is likely for the 0 to +2 oxidation states of most elements in groups 1−12 of the periodic table, for the +1 to +3 oxidation states of group 13 elements, and for the +2 to +4 oxidation states of the group 14 elements. Field shift fractionation and mass-dependent fractionation will often tend to reinforce each other for elements and species in this category, leading to larger observed fractionations. p-, d-, and f-electronic orbitals affect field shifts indirectly; more electrons in these orbitals, and more compact orbital structures, tend to push s-electrons away from the nucleus. So species with more p-, d-, and/or f-electrons will tend to attract larger nuclei. U(IV) vs. U(VI) is an example of this behavior, with the two valence 5f-electrons in U(IV) species leading to higher 238 235 U/ U than in 5f-depleted U(VI) species. This pattern is likely to occur in oxidation states higher than +2 for elements in groups 3–11, lanthanides, and actinides, as well as for the –4 to +2 oxidation states of group 14 elements, the –3 to +3 oxidation states of group 15 elements, the –2 to +4 oxidation sates of group 16 elements, and the –1 to +5 oxidation states of group 17 elements. In these systems, field shift effects and mass-dependent fractionation may tend to oppose each other and partially cancel.

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4. Valence s-orbital electron densities vary much more widely in heavy (high-Z) elements than light (low-Z) elements. So field shift isotope fractionation effects will be much larger for elements with high atomic numbers (Knyazev and Myasoedov 2001; Schauble 2007). Isotopic variation in nuclear charge radii also varies a lot from one isotope pair to another, but it does not show a strong general trend with atomic number (Knyazev and Myasoedov 2001). It is not yet clear what the minimum atomic number needs to be for significant field shift effects to occur. 5. Field shift fractionation factors scale with T–1, whereas equilibrium mass-dependent fractionations tend to scale as T–2 (Eqn. 11). As temperature increases, field shift effects may overwhelm mass-dependent fractionation for some elements.

Mass-independent fractionation signatures in heavy elements, versus light elements In light elements (e.g., oxygen and sulfur), mass-independent fractionations (despite the terminology) ultimately follow from effects of mass differences on reaction rates and photochemical cross sections. In contrast, mass-independent fractionation effects in elements with high atomic numbers appear to be determined mainly by nuclear properties other than mass, including nuclear spin, volume, and shape. This can lead to some confusion, because massindependent phenomena in heavy elements may or may not lead to observable departures from proportionality to isotopic mass differences. Illustrative examples can be found in the thallium and mercury isotope systems. Thallium isotope fractionation in nature is likely dominated by the field shift effect (Rehkämper et al. 2002; Schauble 2007; Nielsen et al. 2015), but there are only two stable thallium isotopes, 203Tl and 205Tl, making observation of mass-disproportionate fractionation impossible in natural samples and impractical in laboratory experiments. Among the four common even-numbered isotopes of mercury (198Hg, 200Hg, 202Hg, and 204Hg), isotope fractionations caused by field shift, magnetic, and mass-dependent isotope effects cannot be distinguished solely on the basis of apparent mass-fractionation relationships, because nuclear volume increases by an almost constant increment with each additional pair of neutrons and the magnetic isotope effect is limited to the odd-numbered isotopes 199Hg and 201Hg.

Ab initio methods for calculating field shift fractionation factors Bigeleisen (1996) and Nomura et al. (1996) identified the nuclear field shift effect in uranium isotope fractionations based on the inverse relationship between 238U/235U and oxidation state, and the close correlation between the magnitude of fractionation in other uranium isotopes and nuclear charge radii variations inferred from optical spectra of uranium vapor. Because of the characteristic pattern of isotopic charge radii, which deviates from pattern of mass differences, they were able to draw conclusions without a quantitative, ab initio theoretical model of the species present in the experiments. Such charge radius pattern matching has been widely used to search for evidence of field shift isotope fractionations in laboratory experiments and natural samples (Fujii et al. 2006a,b, 2009), and it has been useful for ruling out the field shift effect as the main cause of mass-independent signatures in mercury (e.g., Blum et al. 2014). However, it is not well suited to predict how large field shift effects will be in previously unstudied reactions and isotope systems. The ability to make forward models is important in these situations, and may also be necessary in systems where processes that mimic field shift fractionation patterns might be active, such as during nucleosynthesis. Reasonably accurate compilations of nuclear charge radii for almost all stable and long-lived radioactive nuclei are available in the literature (Nadjakov et al. 1994 and updates; Fricke and Heilig 2004; Angeli and Marinova 2013), so the main goal of theoretical models is to determine electron densities in the species of interest. Accurate calculations of electron densities bound to high atomic number nuclei must take account of relativity effects. This can be understood by noting that the kinetic energy of a loosely bound s-orbital electron, when it is momentarily near the center of a highly charged

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nucleus, will be of the same order of magnitude as its rest mass energy. Even the average kinetic energies of inner-shell 1s-orbital electrons are ~100 keV (vs. 511 keV rest mass energy) in elements such as Hg, Tl and U. That implies a velocity that is a significant fraction of the speed of light. For this reason, most ab initio studies of field shift effects to date have been based on the Dirac equation, a relativistic counterpart to the more familiar Schrödinger equation. Some early theoretical studies of field shift fractionation are based on atomic and ionic models of electronic structure (Knyazev and Myasoedov 2001; Abe et al. 2008a). These calculations are easily performed on modern personal computers, but they assume a purely ionic bonding environment. More recently, Dirac–Fock and related model chemistries have been used to directly model molecules (e.g., Schauble 2007; Abe et al. 2008b, 2010; Wiederhold et al. 2010; Fujii et al. 2010). Comparisons to atomic spectra and laboratory measurements indicate that such models are usefully accurate (Schauble 2007; Wiederhold et al. 2010). The basic procedure for constructing a relativistic electronic structure model for calculating field shift fractionation is similar to the initial steps in creating a vibrational model for predicting mass-dependent fractionation, which was outlined in the preceding sections: first an initial structure is selected, and an electronic structure calculation is used to estimate the static energy of the structure; this is often followed up by performing a structural optimization. Vibrational frequencies need not be calculated—if mass-dependent fractionation factors are desired it is usually easier to construct a separate model based on non-relativistic theory, such as conventional DFT. The energy associated with isotope substitution can either be determined using an expression like Equation (24), or (even better) directly by manipulating the size of the nucleus in the first-principles model. Much like Hartree–Fock theory, Dirac–Fock theory has been extended to improve model accuracy by considering electron correlation effects and excited electronic states (see Wiederhold et al. 2010 and Nemoto et al. 2015 for comparisons of model results at various levels of theory). These high-accuracy calculations are much more demanding of computation time and memory, however. Although theoretical studies based on ab initio relativistic electronic structure models have shown good agreement with measurements, the calculations are notably more complex, memory-intensive, and slow than typical non-relativistic methods. As a result, only fairly small molecules (no more than ~20 non-hydrogen atoms) can be modeled easily. Solid and liquid phases must be approximated using small clusters, which is likely to increase model uncertainties. It is difficult to confidently formulate a model of substances with long-range bonding interactions, such as metals or strongly solvated aqueous species, within these constraints. Several different ways around this limitation have been proposed. The first takes advantage of strong correlations between field shift effects in mercury compounds and the effective ionic charges of mercury atoms in those structures. This makes it possible to interpolate fractionations involving complex materials, such as liquid mercury, based on effective ionic charges computed with simpler electronic structure models (Wiederhold et al. 2010; Ghosh et al. 2013). This method may be best suited for group 1, 2 and 12 elements, where the field shift effect is dominated by a single valence s orbital. The second method, introduced by Nemoto et al. (2015), involves the use of a simplified relativistic modeling approach (the Douglas– Kroll–Hess method) that accurately reproduces variations in electron density near high atomic number nuclei with an order of magnitude less computational effort. This raises the possibility of directly modeling larger, more complex molecules while retaining enough accuracy to be useful. The third method, proposed by Schauble (2013) uses fully relativistic Dirac–Fock models (including some electron correlation effects) of simple molecules to calibrate a corresponding set of DFT models that are built using Projector Augmented Wave (PAW) data sets (Blöchl 1994). PAW is closely related to the pseudopotential methods described earlier in this chapter, and is also typically used in conjunction with plane-wave basis sets and periodic boundary conditions. Roughly the same computational effort is required for PAW methods as

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for standard pseudopotential-based DFT. But PAW has the advantage that information about the structure of core electronic orbitals is preserved, so that it is possible to calculate the influence of different chemical bonding environments on electron densities at the nucleus with reasonable accuracy (e.g., Zwanziger 2009). Both PAW and pseudopotential basis sets can be constructed using a partial correction for relativistic effects near the nucleus. Like the simplified relativistic approach proposed by Nemoto et al. (2015), the calibrated PAW method can be applied to larger, more complex materials and molecules, and it can even be applied to metals (such as metallic mercury) and semi-conducting materials where long-range bonding interactions are important. However, the PAW method probably loses some accuracy due to its dependence on a limited calibration set of small molecules.

Isomer shifts from Mössbauer spectroscopy A final method of calculating nuclear field shift fractionations, suggested originally by Knyazev and Myasoedov (2001), uses isomer shifts measured with Mössbauer spectroscopy to determine changes in electron density from one species to another. Isomer shifts and field shifts arise from the same interaction between nuclear charge and electron density, with the main difference being that in Mössbauer spectroscopy the nuclear charge radius changes spontaneously as the Mössbauer nucleus is excited and then decays. In terms of data processing, the isomer shift is distinct from the second-order Doppler shift and NRIXS vibrational spectroscopy—it is typically one of the fundamental parameters measured in a standard Mössbauer experiment (along with quadrupole and magnetic splitting), and does not require measurements at multiple temperatures or a synchotron X-ray source. Like second-order Doppler shift and NRIXS measurements, isomer shifts can be measured in complex materials, selectively and directly probing the Mössbauer isotope’s chemical environment. Knyazev and Myasoedov (2001) showed a promising correlation between calculated electron density variations in vapor-phase neptunium ions with varying charge and measured 237Np-isomer shifts in crystals where Np has the same formal oxidation states. Schauble (2013) went a step farther by comparing 119Sn-isomer shifts with electron densities calculated in the same substances using the DFT-PAW approach. The excellent correlation suggests that Mössbauer spectroscopy will be a powerful tool to predict field shift isotope fractionation in elements with Mössbauer-active isotopes.

CONCLUSIONS While the theory of stable isotope fractionation has been developed in the middle of the twentieth century, the last decade was marked by the growing use of first-principles calculations to apply the theory to non-traditional stable isotopes. The aim of these calculations was first to determine the isotope fractionation factors when the considered phases are in thermodynamic equilibrium in order to identify the factors controlling these equilibrium fractionations. Quantitatively, the theoretical studies mentioned in this chapter but also the others dealing with the traditional stable isotopes show that calculated fractionation factors are reliable enough to be directly compared to experimental values and to values derived from spectroscopic techniques such as Mössbauer and NRIXS. To reach such level of accuracy, high-quality calculations are necessary (the quality of the model can be tested by comparing the calculated structural, electronic and vibrational properties with available experimental data) and most importantly the considered phases must be treated in a consistent manner. In stable isotope geochemistry, first-principles molecular modeling now represents numerical experiments that fully complement laboratory experiments for contributing to the interpretation of isotopic data collected on natural samples. The advances in computation power enable to model systems of increasing complexity. Among the future directions of research that deserve special efforts, we can cite the investigation of isotopic fractionations associated with complex crystal chemical processes

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(e.g., solid solutions, chemical impurities, crystal defects, adsorption complexes), and the exploration of the mechanisms producing mass-independent fractionation, not to mention kinetic fractionation.

ACKNOWLEDGEMENTS This chapter was improved by the thoughtful suggestions from T. Fujii, J. Kubicki and J. Wiederhold.

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Reviews in Mineralogy & Geochemistry Vol. 82 pp. 65-83, 2017 Copyright © Mineralogical Society of America

Equilibrium Fractionation of Non-traditional Stable Isotopes: an Experimental Perspective Anat Shahar, Stephen M. Elardo Geophysical Laboratory Carnegie Institution for Science Washington, DC 20015 USA [email protected]; [email protected]

Catherine A. Macris Indiana University – Purdue University Indianapolis Indianapolis, IN 46202 USA [email protected]

INTRODUCTION In 1986, O’Neil wrote a Reviews in Mineralogy chapter on experimental aspects of isotopic fractionation. He noted that in order to fully understand and interpret the natural variations of light stable isotope ratios in nature, it was essential to know the magnitude and temperature dependence of the isotopic fractionation factor amongst minerals and fluids. At that time it was difficult to imagine that this would become true for the heavier, so called non-traditional stable isotopes, as well. Since the advent of the multiple collector inductively coupled plasma-source mass spectrometer (MC–ICP–MS), natural variations of stable isotope ratios have been found for almost any polyisotopic element measured. Although it has been known that as temperature and mass increase, isotope fractionation decreases very quickly, the MC–ICP–MS has revolutionized the ability of a geochemist to measure very small differences in isotope ratios. It was then that the field of experimental non-traditional stable isotope geochemistry was born. As O’Neil (1986) pointed out there are three ways to obtain isotopic fractionation factors: theoretical calculations, measurements of natural samples with well-known formation conditions, and laboratory calibration studies. This chapter is devoted to explaining the techniques involved with laboratory experiments designed to measure equilibrium isotope fractionation factors as well as the best practices that have been learned. Although experimental petrology has been around for a long time and basic experimental methods have been well-refined, there are additional considerations that must be taken into account when the goal is to measure isotopic compositions at the end of the experiment. It has been only about ten years since these initial studies were published, but much has been learned in that time about how best to conduct experiments aimed at determining equilibrium fractionation factors. We will not focus on the scientific results that have been determined by such experiments, as each chapter in this book will focus on a different element of interest. Instead we will provide a how-to for those interested in conducting these experiments in the future. 1529-6466/17/0082-0003$05.00

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Shahar, Elardo & Macris FACTORS INFLUENCING EQUILIBRIUM FRACTIONATION FACTORS

Equilibrium isotope fractionation is driven by the effects of atomic mass on bond vibrational energy. The relationships are easily understood using a simple molecule as an example. All molecules have a zero-point vibrational energy (ZPE) = ½ hν, where ν is the vibrational frequency and h is Planck’s constant. The vibrational frequency of a particular mode for a molecule can be approximated using Hooke’s Law, ν = ½ π √(k / μ), where k is the force constant and μ is the reduced mass of the molecule (e.g. μ = ma mb / [ma + mb], where ma and mb are the atomic weights of two atoms in a diatomic molecule). When a light isotope in a molecule is substituted by a heavy isotope the potential energy curve does not (to a first approximation) change shape, hence the force constant does not change, but the vibrational frequency does change. When a more massive isotope is substituted, the reduced mass of the molecule increases, which decreases the vibrational frequency and the energy. Therefore, equilibrium stable isotope fractionations are quantum-mechanical effects that depend on the ZPE of the molecule being investigated. The same principle applies to crystalline materials. To a first approximation, the most important factor that determines the magnitude of isotopic fractionation is differences in bond strength; stiffer bonds concentrate the heavy isotope. Bond strength (stiffness) determines vibrational frequency and vibrational frequency determines internal energy. Stiffer bonds tend to correlate with high oxidation state, covalent bonding, and low coordination number. Therefore, all else being equal, the heavy isotopes of an element will partition preferentially into phases with these characteristics. Furthermore, increasing the pressure experienced by a given phase stiffens the bonds while increasing the temperature weakens the bonds. Therefore, the main variables that influence the equilibrium fractionation factor between two phases are temperature, pressure, oxygen fugacity and composition, as these are the variables that will most notably change the bond strength of the isotope of interest. For a thorough explanation of how temperature, pressure, composition and oxygen fugacity affect isotope fractionation the interested reader can reference Young et al. (2002, 2015) and Schauble (2004). It has been known since the work of Urey (1947) and Bigeleisen and Mayer (1947) that temperature is crucial for the determination of equilibrium constants. These seminal works included calculations of equilibrium constants for isotopic exchange reactions as a function of temperature. At high temperature, the equilibrium constant becomes proportional to the inverse square of temperature (i.e., isotope fractionation decreases proportional to 1 / T 2). Thus, at the high temperatures involved in many Earth systems isotope fractionation was considered negligible and ignored for elements heavier than the traditional stable isotopes (C, H, S, O, N). Similarly, the effect of pressure on isotope fractionation was also ignored. Joy and Libby (1960) first calculated the effect of pressure on isotope fractionation. They suggested that oxygen isotope fractionation might be pressure dependent at low temperatures. However, in 1961, Hoering did not observe a pressure effect on oxygen isotope partitioning between water and bicarbonate at 43.5 °C and between 0.1 and 400 MPa. Then in 1975, Clayton et al. found no pressure effect on the calcite-water fractionation at 500 °C, 0.1–2 GPa and at 700 °C, 50–100 MPa. Due to these initial studies, the effect of pressure on isotope fractionation was assumed to be negligible for all elements (except for hydrogen) at all pressures. However, in 1994, a study again predicted that pressure should have an effect on isotopic fractionation (Polyakov and Kharlashina 1994). There are at least two ways in which pressure can affect isotope fractionation: differential molar volume decrease and force constant stiffening. The first cause is due to a molar volume isotope effect in which the heavy isotopes make slightly shorter bonds and therefore pack more tightly than lighter isotopes, and has been discussed in the literature extensively (e.g., Polyakov 1998). This effect comes mainly from the quantum vibrations of the nuclei inside their surrounding electronic cloud. The second cause is due to an increase of the force constants and, correspondingly, of the vibrational frequencies due to the stiffening of the bonds as the volume decreases due to increasing pressure.

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The effect of oxygen fugacity on isotope fractionation was largely untested until the redoxactive transition elements and main group elements could be measured with adequate precision. Iron was the first element to be studied in great detail and since 1999 many studies have shown that, in both low and high temperature environments, the oxidation state (Fe3+, Fe2+) exerts strong control on the magnitude of the Fe isotope fractionation (e.g., Williams et al. 2004; Shahar et al. 2008; Dauphas et al. 2014). Likewise, theoretical and experimental (e.g., Hill et al. 2009) results at low temperature have shown that speciation (bond partner and coordination in a solution) also regulates mass fractionation of Fe. In high-temperature solids or melts, the ligand to which Fe is bound also should affect the Fe isotopic fractionation (Schauble 2004). One of the main reasons that experiments are so critical to the study of equilibrium fractionation factors is that each of these variables can be studied individually in an experiment in order to determine which has the largest effect and how the variables influence each other. As mentioned above, Fe isotopes have been studied in the greatest detail experimentally, so many of the examples in this chapter will focus on Fe. However there are many elements that have been, or have the potential to be experimentally studied.

PROOF OF EQUILIBRIUM IN ISOTOPE EXPERIMENTS The goal of typical stable isotope fractionation experiments is to determine the equilibrium fractionation factor between two phases at a set of temperatures, pressures, oxygen fugacities, and/or compositions of interest. The initial challenge for the experimentalist is to design the simplest, most elegant experiment that will achieve their goal, given the myriad constraints imposed by the laboratory. These issues include (but are not limited to) sample size, time, P–T capabilities of the experimental apparatus, availability and usefulness of appropriate starting materials and oxygen buffers, and the ability to separate phases for analysis after an experiment. For example, experimental apparati usually limit sample sizes to milli- or microgram levels, and realistic laboratory time scales impose challenges with systems involving diffusion as the method of isotope exchange. Additionally, and perhaps most importantly, is the ability to prove that the fractionation measured in an experiment is, in fact, the equilibrium fractionation factor between the phases of interest. This point is nontrivial and must be given significant thought when planning a set of experiments. This section will highlight some of the more effective experimental approaches and techniques employed in the past and speak to best practices for achieving a very close approach to equilibrium in stable isotope fractionation experiments.

Time series The use of a series of experiments conducted over a range of durations to assess the approach to equilibrium is one of the most commonly used methods in experimental petrology. The concept is straightforward: a series of experiments (≥ 3) with the same starting materials is conducted at the same pressure and temperature (usually the lowest of interest) for increasingly longer durations. As the isotopes exchange between phases, the system moves closer to having an equilibrium distribution of isotopes with increasing experimental duration. The equilibrium fractionation between the two phases is estimated by running long enough experiments so that the system appears to have reached a steady state, i.e., there is no further detectable net isotope exchange at progressively longer run times. The run products are then analyzed to ensure that they yield consistent results, which can be used to argue for a close approach to equilibrium (or at minimum a steady state). The time series results then guide future experiments. A time series should also be used to assess chemical equilibrium independent of isotopic equilibrium. Chemical zoning in experimentally grown phases is always an indicator of disequilibrium and should lead to longer duration experiments whenever practical. An exception to this is dynamic cooling and/or decompression experiments designed to assess kinetic effects at controlled cooling and/or decompression rates. Some chemical zoning is

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tolerable in more traditional phase equilibrium experiments, as microbeam analyses can be made at phase boundaries where a local equilibrium may exist. However, most isotopic fractionation experiments require the physical separation of phases and analysis of the bulk phase. In this case chemical zoning is less tolerable as it will likely be accompanied by isotopic zoning. For this reason, the durations of phase equilibrium experiments designed to investigate isotopic fractionation may need to be longer than is typical for studies of element partitioning or the like (Shahar et al. 2008). In situ isotopic analyses using laser ablation MC–ICP–MS (LA–MC–ICP–MS) or secondary ion mass spectrometry (SIMS) are ways to determine if isotopic zoning has occurred in an experiment, and to obtain spatially oriented measurements of the experimental product. However, extreme care must be taken when treating the resulting data to show the measured values represent a close approach to equilibrium.

Multi-direction approach The multi-direction approach is also discussed in O’Neil (1986) and has been termed “reverse reactions” in the experimental petrology community (Fig. 1). It is an extension of the time series described above with the added benefit of approaching the equilibrium isotope value from two sides, which results in a more precise determination of the equilibrium value by bracketing. This is done by conducting at least two time series in which the difference between the isotopic compositions of the starting materials lie on either side of the equilibrium isotope fractionation value of the system (line labeled ‘Equilibrium’ in Fig. 1). Over time, the two sets of starting materials (differing only in isotopic value) evolve towards the equilibrium isotopic state of the system, thereby bracketing the true equilibrium fractionation factor of the two phases. Although this idea seems straightforward there are subtleties associated with non-ideal conditions and solid phases that could complicate the system which are discussed in great detail by Pattison (1994). Although the multi-direction approach is more rigorous than a simple time series, its accuracy in determining the equilibrium fractionation factor is limited by the ability to come close to equilibrium in laboratory time scales. For some systems, this technique may not be practical due to sluggish isotope exchange. In such cases, a different strategy is required to determine equilibrium fractionation factors. t0

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Figure 1. A schematic of a multi-direction approach experiment showing two phases (‘A’ and ‘B’) that have starting fractionation values (DiEA-B) lying on opposite sides of the equilibrium value. As the experimental duration progresses the values bracket the true equilibrium value. Here ‘i’ represents an arbitrary isotope of element ‘E’, and DiEA-B = diEA - diEB.

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The Northrop–Clayton method is a variation of the multi-direction approach that allows an estimation of the equilibrium fractionation between two phases by extrapolation. This is a strategy that can be employed when attaining equilibrium is not possible on time scales relevant to laboratory experiments. This approach, termed the “partial exchange technique” in O’Neil (1986), was first described by Northrop and Clayton (1966) and later by Deines and Eggler (2009). The partial exchange method involves two phases that each contain at least one site for an element of interest that exchanges isotopes throughout an experiment. A series of experiments are conducted with identical solids and several isotopically variable fluids that bracket the equilibrium isotopic fractionation factor. Just like in the multi-direction approach, the evolution of the fractionation with time will bracket the equilibrium value. The benefit of the Northrop–Clayton method is that a close approach to equilibrium is not required to estimate the fractionation factor. This is true because this method assumes that rates of isotopic exchange for ‘companion’ experiments (runs that are the same in every respect except the isotopic composition of starting materials) are identical. This assumption allows one to extrapolate to the equilibrium value of the system based on the ‘percentage of isotopic exchange’ in two or more companion experiments (O’Neil 1986). For this method, the extent of isotopic exchange is directly proportional to the accuracy of the fractionation factor. In other words, if the isotopes are not largely exchanged, the fractionation might appear to be larger than the true equilibrium value. A combination of this partial exchange technique and the classical time series approach was used successfully by Schuessler et al. (2007) in obtaining the equilibrium Fe isotope fractionation between pyrrhotite and rhyolitic melt.

Three-isotope exchange method Another method of determining equilibrium fractionation factors is termed the threeisotope exchange method. This method was pioneered at the University of Chicago (Matsuhisa et al. 1978; Matthews et al. 1983) for oxygen isotopes between a mineral and an aqueous phase and later modified by Shahar et al. (2008) to determine direct mineral–mineral fractionation of Fe isotopes. This method requires that the element of interest has at least three stable isotopes in measurable abundances and utilizes the addition of a known amount of isotopic ‘spike’ (addition of a specific isotope to a phase in excess of its natural abundance) to one of the starting materials containing the element of interest. The choice of which isotope to use as the spike is made obvious by looking at a three-isotope plot (Fig. 2). By adding an excess amount of the isotope present in the denominator of both axes (represented by an ‘x’ in the axes labels on Fig. 2A, B) to one of the starting materials, the spiked phase will be displaced from the terrestrial fractionation line (TFL) in three-isotope space by a line with a slope of unity. The principle of the three-isotope exchange method is to replace the TFL, which has a zero intercept on a three-isotope plot, with a secondary fractionation line (SFL) with the same slope as the TFL, but a non-zero intercept determined by the bulk composition of the isotopically spiked system. Figure 2A depicts the typical trend to equilibrium expected in these experiments. As the spiked (Phase 2 Initial) and unspiked (Phase 1 Initial) starting materials exchange isotopes at the pressure and temperature of interest for increasingly long times (t1, t2), their isotopic compositions migrate towards the SFL. When isotope exchange is complete and the system reaches isotopic equilibrium, all phases containing the element of interest will lie on the SFL (Phase 1 & 2 Equilibrium), and the equilibrium fractionation factor will be defined by the distance between the final measured values along the x- and y-axes corresponding to isotopes ‘j’ and ‘i’, respectively. The three-isotope exchange method can also be used to extrapolate to the equilibrium isotopic value in cases where exchange is slow. In this case it is quite important that the experiment not be ‘over’ spiked so as to limit the amount of extrapolation that is necessary. A good rule of thumb is to incorporate no more (and sometimes much less) than 1% of the element as a spike.

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Phase 1 Initial

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Figure 2. A schematic of the three-isotope exchange method shown for arbitrary isotopes ‘i’, ‘j’, and ‘x’, of element ‘E’. Figure 2A depicts the more traditional path during a dissolution and precipitation experiment. Phase 1 (red circles) is unspiked and starts on the terrestrial fractionation line. Over time, as the system evolves towards equilibrium, Phase 1 moves to lighter values, while Phase 2 (blue squares), which is spiked so that is starts off of the terrestrial fractionation line, moves towards heavier values. The yellow star represents the bulk value of the system. Equilibrium is reached when both phases reach the secondary fractionation line, though in some cases the phases may not fully reach the equilibrium values, as depicted by the transparent symbols. In such cases, the equilibrium fractionation is determined by extrapolation. Figure 2B depicts a higher temperature experiment where all the isotopes first mix at the bulk value for the system and then ‘unmix’ along the secondary fractionation line. In this scenario a time series is also needed to prove equilibrium.

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By way of practical example, we turn to the three Fe isotopes, a system for which the three-isotope method has been executed successfully for mineral–mineral, mineral–fluid, and metal–silicate exchange experiments (e.g., Shahar et al. 2008, 2015; Beard et al. 2010). For Fe, the axes of the three-isotope plot are defined by two isotope ratios with the same denominator: 56Fe/54Fe on the y-axis and 57Fe/54Fe on the x-axis (see Fig. 2 of Shahar et al. 2008). These ratios are compared to those of a standard and reported in per mil (‰) using the conventional delta notation:

( (

) )

 i Fe/ 54 Fe  Sample  di Fe =103  i 1  Fe/ 54 Fe  Standard   where i = 56 or 57. If the goal is to derive the equilibrium isotope distribution by extrapolation, one can take advantage of the known equilibrium mass fractionation relationship between the two isotopes of interest: γ

 103 + d57 Fe  3 d56 Fe = (103 + d56 Fe bulk )  3  - 10 57 + d 10 Fe bulk  

This slightly concave fractionation relationship can be approximated by a straight line in three-isotope space with a slope of approximately γ and an intercept defined by the bulk isotopic composition. The intersections of the line defined by the above equation for the SFL and the two lines derived from the trajectories of the isotopically evolving starting materials represent extrapolation to the equilibrium values of the system. The exponent, γ, is 0.67795 (for Fe) at equilibrium based on the equation for equilibrium mass fractionation: 1 1 m56 m54 γ= 1 1 m57 m54

where m is the mass of subscripted Fe isotope. Although both methods offer extrapolation to equilibrium, an advantage of the three-isotope technique over the Northrop–Clayton method is the ability to know when isotopic exchange is complete and equilibrium has been reached, i.e., when the final isotope values lie on the SFL. An entirely different path towards equilibrium using the three-exchange isotope method is presented schematically in Figure 2B. Here, the spiked (Phase 2 Initial, t0) and unspiked (Phase 1 Initial, t0) starting materials are thoroughly chemically and isotopically mixed before equilibrating because either the sample is completely melted or dissolved during the experiment. When the system homogenizes at the beginning of the experiment (t1), the isotopes mix and collapse to the bulk isotopic value of the system (yellow star), which by definition lies on the secondary fractionation line (SFL). Following mixing (t1), there is an ‘unmixing’ (t2, t3) among the isotopes along the secondary fractionation line (SFL) as the system equilibrates isotopically as well as chemically over time. In progressively longer experiments, the isotopic values of the phases move away from each other along the SFL until they reach their equilibrium values (Phase 1 & 2 Equilibrium, t4). In this case, equilibrium is determined by time series during unmixing; i.e., when the values stop changing (reach a steady state) along the SFL, the system is thought to be in equilibrium and the fractionation factors can be determined. This is critically important and considered a rigorous way to prove equilibrium in these experiments.

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The utility of the three-isotope method is in the ability to trace the trajectories of isotopic exchange between phases in three-isotope space as the experiments progress through a time series. Without an isotope tracer (the isotopic spike) it could be difficult to prove that isotopic equilibrium had been reached, know whether the different phases had exchanged isotopes, and be certain that the exchange occurred in a closed system. By spiking one of the phases in the experiments with a known amount of spike, there is better control on mass balance and one can determine if the experiment was truly a closed system. Open system behavior was successfully diagnosed by Lazar et al. (2012) in three-isotope exchange experiments involving Ni alloying with Au capsules. Proving that the isotopes have come to equilibrium is a crucial part of conducting high temperature experiments. If the isotopes are mixed but have not equilibrated yet then the fractionation would either be underestimated or determined as zero. In silicate–metal experiments, the majority of the Fe in the experiment is in the metallic phase, so any contamination from the metal to the silicate (which cannot be seen by the eye) will cause the apparent fractionation factor to be underestimated or appear to be zero. However, with the three-isotope exchange method this can be easily seen and a duplicate experiment would be performed and more carefully separated to obtain the correct result. Also, it is important to remember that chemical equilibrium is not equivalent to isotopic equilibrium and must be independently assessed. Shahar et al. (2008) found that while chemical equilibrium between fayalite and magnetite occurred within 6 hours, isotopic equilibrium required 48 hours at 800 °C. This is extremely important for experiments where equilibrium is assumed to have occurred based on electron probe measurements showing equilibrium textures and chemical compositions. The use of a classical time series to assess the approach to equilibrium in isotopic fractionation experiments is valuable even in conjunction with other methods and we highly recommend its use in all experimental studies of isotope fractionation. Furthermore, the three-isotope exchange method may not be available for use in all systems of interest. In such cases, the use of a time series is essential and is highly recommended in conjunction with a multi-directional approach. Frierdich et al. (2014) argued that even the three-isotope technique is not sufficient on its own to guarantee that there are no possible kinetic contributions to the data. The authors argued that adding a multi-direction approach to the three-isotope technique unambiguously demonstrates that equilibrium has been reached during the experiment. Investigators should thoughtfully develop an experimental plan involving some combination of the techniques presented above that will most rigorously yield the equilibrium isotope fractionation factors sought for their system.

Kinetic effects Kinetic isotope effects are commonly observed in nature and in the lab, making it crucial to be able to distinguish between isotopic equilibrium and disequilibrium. Both states are important to understand and quantify for isotopic systems, but the experimentalist must be able to discern which of the two is represented in their experiments. Kinetic isotope effects are usually associated with fast, unidirectional, or incomplete processes (O’Neil 1986). Some of the pathways that might lead to kinetic effects in isotope exchange experiments are evaporation, diffusion, and rapid crystallization. No matter how carefully an experiment is designed there is always the possibility of a kinetic isotope effect that has not been considered. From quenching effects in a piston cylinder to isotopic zoning within crystals, it is nearly impossible to avoid all kinetic isotope effects. However, this does not mean that the equilibrium value cannot be determined if the kinetic processes are properly identified and understood, and the data are treated accordingly. For example, Shahar et al. (2011) noted that when measuring Si isotopes in situ by laser ablation MC–ICP–MS, the isotopic values within each phase showed small deviations thought to be due to a kinetic effect during the quenching of the experiment. In order to determine the true isotopic ratio of the phases, the values were averaged. Had the experiment been measured by solution instead of by laser ablation these local variations would not have been seen and the kinetic behavior would have gone unnoticed.

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In some cases kinetic processes are not an impedance to equilibrium, but a means to achieve it. For example, diffusion may be the dominant mechanism by which isotopes exchange between phases in an experiment. In this case, it is important to understand the rates of diffusion corresponding to the isotopes and materials of interest. Diffusion coefficients for many systems have been experimentally determined and can be used in some cases to make predictions as to how long it should take for two phases to chemically and isotopically equilibrate at the pressure and temperature of interest. This information can be used to guide the experimentalist when deciding on experimental durations in a time series. In an experiment depending on diffusion (or another kinetic process) to exchange isotopes, it is critically important to allow enough time for a steady state to be reached so that a reasonable estimation of equilibrium can be made. However, it is also possible to achieve a spurious result by running an experiment for too long. For example, Lazar et al. (2012) suggested that there is a divergence in the isotope ratios after an experiment had reached equilibrium when kinetic effects can start to play a role again, especially when there is diffusive loss to the capsule. Therefore it is important to remember that while a time series is a good way to prove that isotopic equilibrium has been reached, there is a limit to the duration of the experiment that should not be passed. These are just two of many examples of how kinetic effects can be found within experimental charges even when isotopic equilibrium has been reached.

EXPERIMENTAL METHODS Since the advent of MC–ICP–MS and associated chemical purification methods that allow for the accurate measurement of small isotopic fractionations between co-existing materials, researchers have been exploring pressure, temperature, and composition space in non-traditional stable isotope exchange experiments. Drawing from the well-established experimental methods used successfully by geochemists and petrologists for decades, as well as newer methods developed by mineral physicists and beam line scientists, the community is making great advances in our understanding of how non-traditional stable isotopes fractionate in geo- and biogeochemical processes. In order to reproduce the range of conditions occurring from Earth’s surface to its core, researchers are utilizing a variety of experimental apparatuses, including simple flasks and low-temperature ovens, cold-seal hydrothermal vessels, piston– cylinders, multi-anvils, and diamond anvil cells. In the following sections we provide a brief (non-exhaustive) review of these methods, starting with low temperature experiments and their applications, and then moving progressively higher in temperature and pressure.

Low temperature experiments Early experiments on the non-traditional stable isotopes focused on low temperature Fe isotope fractionation in aqueous fluids (e.g., Johnson et al. 2002; Welch et al. 2003; Hill et al. 2009) and more commonly between an aqueous fluid and a mineral (e.g., Johnson et al. 2002; Skulan et al. 2002). At this time in Fe isotope geochemistry history, it was unclear if abiotic systems could fractionate isotopes as efficiently as biologic systems, so a baseline was needed in order to determine if Fe isotopes could be used as a biosignature. These experiments employed the use of an isotopic spike (excess 57Fe) in combination with classical time series and multi-direction approaches to assess Fe isotope fractionation between phases at low temperatures. Johnson et al. (2002) and Welch et al. (2003) investigated fractionation in aqueous Fe(II) and Fe(III) solutions at room temperature and in an ice bath. These studies have implications for the fractionation of Fe isotopes between aquo and hydroxy complexes in oxygenated natural waters.

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If isotope fractionation associated with a redox change exists it would be most noticeable at low temperature and amongst two phases with strikingly different bonding environments, such as a fluid and a solid. As explained in detail in Skulan et al. (2002), low-temperature mineral–fluid exchange experiments rely on a steady state of dissolution and precipitation so that the isotopes can move around within the experiment and tend towards equilibration. This study utilized a range of experimental strategies to ultimately distinguish kinetic effects from equilibrium fractionation between hematite and aqueous Fe(III). One approach they used was simply mineral synthesis. The premise is that a mineral, such as hematite, will be thermodynamically stable at the conditions of the experiment, but will not be present in the starting materials, thus facilitating precipitation. In this case, Skulan et al. (2002) reacted aqueous Fe(III) with dilute HNO3 in sealed Pyrex flasks at 98 °C to precipitate hematite from solution. The first hematite crystals will not be in isotopic equilibrium, but as the experiment progresses the system will reach a steady state of dissolution and precipitation in which the hematite crystals will then record the equilibrium fractionation between the crystal and the fluid. The main disadvantage to these experiments is that at low temperature it is much more difficult to attain isotopic equilibrium as the rates of diffusion and nucleation are slow. There are also several kinetic effects that must be sorted out such as isotopic inhomogeneity in the mineral (i.e., zoning during precipitation) or other kinetic effects associated with the rates of dissolution and precipitation. In order to speed up diffusion rates, hydrothermal experiments can also be done using a cold seal technique at slightly elevated temperatures. For example, Li et al. (2015) determined the equilibrium Mg isotope fractionation between dolomite and aqueous Mg in multi-direction mineral synthesis experiments using a 25Mg spike to track isotope exchange. These experiments were conducted at 130–220 °C using Parr bombs and an internal sealed container. By using this technique the experimentalist can avoid possible kinetic effects due to energy barriers that could inhibit crystal growth or dissolution. Another subject of interest in low temperature experimental work is the isotopic effect of adsorption onto mineral surfaces (e.g., Barling and Anbar 2004; Icopini et al. 2004; Johnson et al. 2004; Beard et al. 2010; Nakada et al. 2013; Wasylenki et al. 2014, 2015). These experiments are aimed at understanding the isotope fractionation between aqueous metal species and sorbed metal complexes. In many of these experiments it has been shown that the isotope fractionation is constant throughout the experiment even as the amount of the element sorbed increases. To prove isotopic equilibrium in these experiments a time series and/or the three-isotope technique has been used; we recommend the use of both whenever possible. Additionally, in these types of experiments, the fractionation of interest in nature might be due to kinetic processes or opensystem equilibrium-driven fractionation. Therefore, the approach to treating and interpreting the data may include modeling by constant offset as a function of reactant/product ratio (equilibrium closed-system), or as Rayleigh behavior (equilibrium open-system, or kinetic). For a more thorough explanation of these different treatments, see Johnson et al. (2004).

High temperature, low pressure experiments The least amount of experimental work to date has focused on experiments at hightemperature and ambient or low pressure (≤ 0.5 GPa). This is due in part to the assumption that pressure was not an important parameter in isotope fractionation, and the more practical concern that it is typically easier to achieve a closed system when conducting high-pressure experiments. However, there have been a few key studies conducted in low-pressure experimental apparatuses in order to study the effect of oxygen fugacity and temperature on isotope fractionation. As far as we are aware, the first experiment on fractionation between a silicate melt and metal was conducted in this fashion in a 1 atm gas-mixing furnace (Roskosz et al. 2006). Figure 3A depicts a cutaway of the interior of such a furnace, exposing a molten sample suspended by a metal wire loop (e.g., Pt, Re) in a sealed alumina or mullite tube, which is flushed with an appropriate gas mixture (e.g., CO–CO2, CO2–H2) to impose the chosen oxygen fugacity.

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Figure 3. Schematics of A) a controlled-atmosphere metal loop experiment, B) a piston cylinder experiment (after Young et al. 2015), C) a multi-anvil experiment (after Bennett et al. 2015) and D) an NRIXS experiment (after Dauphas et al. 2012). A) A cross-sectional view of the internal arrangement of a typical ambient pressure controlledatmosphere wire loop experiment. Typical gas mixtures consist of CO–CO2 and H2–CO2, though other mixtures are sometimes used. Wire loops are typically Pt or Re. B) A cross-sectional view of a typical piston cylinder experiment showing where the capsule resides and typical assembly materials. C) An external view of a typical multi-anvil ceramic octahedron and a cross-sectional view of the internal arrangement of the octahedron. D) A typical NRIXS experimental set-up. Incoming synchrotron X-ray pulses (separated by 153 ns) first pass through the diamond premonochromator (PM), and then through the Si channel-cut high resolution monochromator (HRM). The beam is focused by small horizontal (20 cm) and vertical (10 cm) Kirkpatrick–Baez (KB) mirrors onto the DACTA sample. The forward synchrotron Mössbauer spectroscopy (SMS) signal is collected through the diamond axis on an avalanche photodiode (APD). Three additional side APDs collect the NRIXS signal through the X-ray transparent gasket.

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In the study by Roskosz et al. (2006), it was shown that at 1500 °C there was a resolvable and quite large fractionation of Fe isotopes between silicate melt and metal in a series of Pt loop experiments, although it proved difficult to demonstrate isotopic equilibrium. These experiments used the propensity for Fe to alloy with Pt as a way to segregate the metal from the silicate at a fixed f O2 and temperature. The authors noted that kinetic fractionation was rampant during the first ~ 300 minutes of an experiment. Initially, the Fe alloying with the Pt was isotopically light compared to that in the silicate melt and became heavier as the experiment approached equilibrium. In the longest duration experiment, the relative fractionation flipped such that the silicate was lighter than the metal, thus illustrating the need for, and utility of, conducting a time series in such experiments. Another low-pressure study by Schuessler et al. (2007) used an internally heated pressure vessel (IHPV) to investigate Fe isotope fractionation between a rhyolitic melt and pyrrhotite between 840 and 1000 °C at 0.5 GPa. A rigorous assessment of the approach to equilibrium was employed in this study. Three sets of experiments were done to distinguish kinetic effects from equilibrium fractionation for the system. The first was a classical time series in which 57Fe spiked glass was reacted with natural pyrrhotite to determine the time required for a close approach to equilibrium. The second set of experiments employed the Northrop–Clayton method of partial isotope exchange discussed above to extrapolate to the equilibrium value (Northrop and Clayton 1966). In the third set, pyrrhotite was crystallized from the melt instead of being present in the starting materials. This combination of experimental approaches essentially constituted a multi-direction assessment of equilibrium supported by time series. Other challenges for low-pressure experiments may include (but are not limited to) the effects of slow diffusion creating chemical and isotopic zoning in solid mineral or metal phases and difficulty in obtaining pure phase separates post-experiment. We recommend that the interested investigator draw upon the extensive literature regarding the use of experimental techniques such as controlled-cooling rates (e.g., Donaldson 1976; Corrigan 1982), temperature cycling (e.g., Mills and Glazner 2013), and crystal seeding (e.g., Larsen 2005; Dalou et al. 2012) to develop methodology to overcome these issues. Further isotope work using a controlled-atmosphere furnace or IHPV will need to build on the pioneering work of Roskosz et al. (2006) and Schuessler et al. (2007), as these types of experiments potentially have huge advantages over high pressure experiments, such as long-term temperature stability, large sample volumes, and precise control over oxygen fugacity.

High temperature and pressure experiments Many experiments to date have been carried out in a piston cylinder apparatus (Fig. 3B) and a few in the multi-anvil apparatus (Fig. 3C). Both of these instruments allow the experimental materials to stay in a sealed container while at the pressure and temperature of interest throughout the experiment. The benefit is that at elevated pressures and temperatures, equilibrium is more easily attained, and with closed capsules, the composition is less likely to change throughout the experiment. Most experiments are carried out using a ½-inch piston–cylinder cell assembly (Fig. 3B), which is typically capable of maintaining pressures up to ~ 4 GPa. The cell assembly varies by laboratory but a typical assembly, as illustrated by Figure 3B, consists of a graphite tube heater surrounded by a pressure medium, the most common of which are BaCO3, talc or NaCl with an inner Pyrex sleeve, and occasionally CaF2. A thermocouple is inserted axially into the cell through a small hole in the stainless steel base plug, which is typically surrounded by a thin insulating pyrophyllite or Pyrex sleeve. Capsule material also varies according to the specific needs of the experiment (e.g., graphite, Au, Pt), and capsules are typically centered within the cell. Hotspots are typically large (i.e., on the order of millimeters) in piston–cylinder experiments, so the thermal gradient over a sample is not expected to exceed ~ 15 °C in most cases. The first experiment to use the piston–cylinder apparatus to directly determine

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mineral–mineral fractionation was a precipitation–dissolution experiment to measure the Fe isotope fractionation between fayalite and magnetite (Shahar et al. 2008). Multi-anvil cells have smaller sample sizes but can extend the accessible pressure range to approximately 25 GPa (Fig. 3C). Multi-anvil assemblies typically consist of a ceramic octahedron with a cylindrical sample chamber drilled through the center. A heater, typically made of graphite, Re-foil, or La-chromite surrounds the sample capsule and confining spacers. A thermocouple can be inserted either axially or radially depending on the assembly. Although the multi-anvil apparatus can access a far greater pressure range than a piston–cylinder, there is a significant tradeoff with sample volume. The maximum pressure achievable by a particular multi-anvil cell assembly is inversely proportional to the maximum sample size within that assembly. For an in-depth explanation and visualization of piston–cylinder and multi-anvil experimental setups, the interested reader is referred to Bennett et al. (2015). One of the most important choices an experimentalist has to make is what type of capsule to use in an experiment. The choice of capsule material is crucial to the success of the experiment, as in order to assess the approach to isotopic equilibrium and ensure that the possibility of kinetic processes is limited, a closed system must be maintained. An open system, with respect to isotopes, will in all likelihood result in non-equilibrium conditions during the experiment. Therefore, a capsule that alloys or reacts with the element of interest should not be used for the experiment (e.g., metal capsules in metal–silicate experiments simulating core formation). This would lead to a loss of the element of interest into the capsule or the creation of a mineral on the capsule wall that has an isotope fractionation of its own (possibly kinetic) and would therefore change the mass balance of the system and the fractionation throughout the experiment. There is no perfect capsule material in experimental petrology, but acceptable choices can be found. Graphite capsules are the most popular choice. Graphite acts as an inert phase in most systems and is well-suited for many investigations, provided that the condition of carbon saturation can be tolerated. Noble metal and alloy capsules are another popular choice. Platinum, Re, Au, Ag, and Au–Pd are the most commonly used metal capsules; each having its own strengths and drawbacks. The most popular metal capsule, Pt, will cause Fe depletion from silicate melts (Merrill and Wyllie 1973). Although this behavior is well-characterized in the literature and such capsules can be pre-saturated with Fe to minimize this issue, the Pt will provide an additional “sink” for Fe that must come to isotopic equilibrium with the melt in order to avoid kinetic fractionations. Other potential capsule materials include olivine, magnesia, alumina, boron nitride, and silica glass/fused quartz. When the parameters of an experiment require a compromise on capsule material, it is best to test the conditions of interest with more than one capsule material to confirm the results. Isotope fractionation experiments present an additional challenge over more traditional high P–T experiments in that two phases of interest typically need to be physically separated and analyzed in solution by MC–ICP–MS. As the achievable pressure range of an experimental apparatus increases, sample sizes necessarily decrease and the ability of a researcher to truly separate the phases, purify the element of interest, and obtain a precise isotope ratio becomes more limited. A potential work around is to use a selective leaching process (Williams et al. 2012) to chemically separate the phases of interest. However, the possibility of isotopic fractionation during the leaching process exists. For this reason though there are not as many published studies of multi-anvil experiments as piston–cylinder experiments. However, it is expected that more researchers will push the current boundaries with this technique, especially if using a large volume multi-anvil assembly. Selective leaching (also called preferential dissolution) as a strategy to separate phases post-experiment was also used successfully by Macris et al. (2013). This study determined the equilibrium Mg isotope fractionation between spinel and forsterite in piston–cylinder

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experiments at 1 GPa and 600–800 °C. Two Au capsules were placed side by side in the cell assembly, one containing spinel spiked with excess 24Mg and ‘normal’ magnesite, and the other containing forsterite spiked with excess 24Mg and ‘normal’ magnesite. The three-isotope technique was used to determine the spinel–magnesite fractionation and the forsterite-magnesite fractionation, then the spinel–forsterite fractionation was obtained by difference. The magnesite in these experiments served two purposes: it acted as the Mg isotope exchange medium and it allowed for complete separation of phases after the experiment by preferential dissolution. The use of carbonate as an exchange medium has been well known since the pioneering oxygen isotope fractionation experiments of Clayton et al. (1989) and Chiba et al. (1989). It is thought to be superior to water as an isotopic exchange medium and reference material because of potential ‘isotope salt effects’ that can affect the observed fractionation in an experiment due to mineral dissolution changing the fluid composition (Hu and Clayton 2003). Experiments in the piston–cylinder have relied on a time series approach or utilized the three-isotope technique to address isotopic equilibrium. Whenever possible both methods should be used to demonstrate a close approach to equilibrium. The Fe isotope system has been the most-well studied system thus far with the piston–cylinder (Poitrasson et al. 2009; Hin et al. 2012; Shahar et al. 2015). However the piston cylinder has proven useful for studies of many other isotope systems such as S (Labidi et al. 2016), Mo (Hin et al. 2013), Cr (Bonnand et al. 2016), Si (Shahar et al. 2009; Kempl et al. 2013; Hin et al. 2014), Mg (Richter et al. 2008; Huang et al. 2009; Macris et al. 2013), Ni (Lazar et al. 2012) and more. Multi-anvil experiments have been conducted mostly for Fe isotopes (Poitrasson et al. 2009; Williams et al. 2012), but also for Si isotopes (Shahar et al. 2011). It is our belief that these experiments will begin to become more commonplace, leading to a more complete database of experimentally determined equilibrium fractionation factors.

NRIXS and diamond anvil cell experiments One might imagine that if multi-anvil experiments have sample sizes that are almost too small to be useful, then diamond anvil cell experiments would be all but impossible. However, it is important to know if pressure truly is a negligible variable within isotope geochemistry (e.g., Clayton et al. 1975). Therefore, a new technique has emerged in order to measure isotope fractionation factors at high pressure. Polyakov and co-workers (Polyakov et al. 2005, 2007; Polyakov 2009) have pioneered the use of nuclear resonant inelastic X-ray scattering (NRIXS) synchrotron data to obtain vibrational properties of minerals from which isotopic fractionation factors can be calculated. The technique is based on the fact that certain isotopes possess a low-lying nuclear excited state that can be populated by X-ray photons of a particular energy and therefore directly probe the vibrational properties of the solid. As shown in Figure 3D, in nuclear resonant X-ray spectroscopy an in-line high-resolution monochromator is used to narrow down the photon energy to meV resolution and fine-tune the monochromatic X-ray near the exceedingly narrow nuclear resonant (elastic) line. Avalanche photodiodes (APD) are used to collect only the signal from nuclear resonance absorption and reject non-resonantly scattered radiation. The APD directly downstream of the sample collects nuclear resonant forward scattering, or synchrotron Mössbauer spectroscopy (SMS) signal, which can provide a precise determination of the hyperfine interaction parameter and Lamb–Mössbauer factor. The APDs surrounding the sample radially collect the NRIXS signal, which is a result of creation (Stokes) or annihilation (anti-Stokes) of phonons as the incident X-ray beam is scanned over a small range (approximately ±100 meV) around the resonant energy. The phonon density of states (DOS) is extracted from this phonon excitation spectrum associated with the nuclear resonant isotope (Hu et al. 2003). The integration of the whole DOS provides values for key thermodynamic properties. In addition, NRIXS data can also be used to derive reduced partition function ratios (β-factors) from which equilibrium isotopic

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fractionation factors can be calculated: δA – δB = 1000 × (ln βA – ln βB), where A and B are two different phases of interest. This technique has been widely used by the mineral physics community to investigate seismic velocities and phonon density of states of high-pressure minerals (Sturhahn et al. 1995) but is relatively new to the isotope geochemistry community. The most commonly used software for processing NRIXS data is the PHOENIX program. The PHOENIX program (Sturhahn 2000) analyzes the data obtained by NRIXS and exports three different force constants. The first is the determination of the force constant using the moments approach in which the normalized excitation probability is computed using the raw data and the experimental resolution function. The outcome is based on the raw data and therefore multiphonon contributions are included in the analysis. The second determination of the force constant is also based on this moments approach; however it is based on data that are first ‘refined,’ that is, the multiphonon contribution is corrected for and S(E) is extrapolated. Dauphas et al. (2012) argued that this moments approach is the best method for determining the force constant as it has smaller uncertainties, is not as dependent on the background, and is not as sensitive to asymmetric scans. The third determination of the force constant is based on the partial density of states that is computed from the raw data. The force constant is then calculated based on the density of states and is therefore only based on the one phonon contribution of the data and does not need to be corrected for the possible multiple phonon contributions in the data. Murphy et al. (2013) suggested that this last force constant calculation is the most robust. The two methods produce slightly different force constants thereby changing the beta factor calculation as well. Dauphas et al. (2012) also suggested that the high-pressure data used by Polyakov (2009) were inaccurate due to the truncation of the high-energy tail during the data acquisition. These data were originally collected by different groups to derive seismic velocities and were not specifically intended for the novel applications pioneered by Polyakov (2009) regarding β-factors. However, since this type of data has proven effective in calculating equilibrium isotope fractionation factors, its collection and treatment has been improved. For example, Dauphas et al. (2012) further suggested that longer acquisition times are necessary to determine the partial phonon density of states (PDOS) of the phases at high pressure with particular emphasis on the high energy tail. To aid in this, Dauphas et al. (2014) introduced a new program, SciPhon, which determines the background on each side of the elastic peak and subtracts them individually instead of using one background value for the whole spectral range. The broadest spectral energy ranges allow for the most accurate determination of the background correction, apparently reducing overall uncertainty. A comparison of the force constant obtained from PHOENIX and SciPhon indicate that SciPhon uses more accurate background subtraction, gives values closer to theoretical calculations, and is more consistent over differing energy ranges (Shahar et al. 2016). Therefore, SciPhon should be used to calculate force constants when conducting NRIXS experiments. Roskosz et al. (2015) successfully applied SciPhon to NRIXS data to describe the equilibrium Fe isotope fractionation between spinels and silicates, which is in agreement with the direct experimental determination by Shahar et al. (2008). Once the force constant is calculated, the β-factor determination is straightforward (e.g., Polyakov 1998; Murphy et al. 2013). These calculations are then used to obtain an estimate of the equilibrium isotope fractionation as described above: the β-factors are calculated for each phase that is being probed and then subtracted from one another to get the isotopic fractionation between the two phases. In this way, each phase is analyzed in situ and does not need to be separated postexperiment. To date, this technique has been used solely for room temperature experiments, both at atmospheric pressure and high pressure (Dauphas et al. 2012; Shahar et al. 2016). It is expected that high temperature experiments will follow when the temperature gradients in the diamond anvil cell can be better controlled and reduce uncertainties in the analyses.

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The characterization of run products for experiments designed to investigate isotopic fractionations is typically more in-depth than for traditional petrologic experiments, and in most instances requires more sample mass. For low-temperature experiments, both the phases involved and experimental techniques employed are more varied, and are often specific to the system being investigated. Here we will focus on the process for hightemperature experiments by way of example. Ambient-pressure, piston–cylinder, and multi-anvil experimental run products are all similar in their analytical requirements. Once the samples have been removed from the experimental assembly, the samples are broken or cut in half. For piston–cylinder and especially multi-anvil assemblies, it is best practice to cut the sample capsules axially rather than radially in order to assess any compositional gradients that may be present due to temperature gradients in the cell. The first portion of the run product is mounted and polished for electron microprobe analysis. Geochemical characterization of the run product via electron microbeam analyses is essential and should always be conducted in tandem with isotopic analyses. The rest of the sample is then processed for isotopic analyses. The most effective method for phase separation will likely vary with the phases of interest. Fortunately, in the case of experiments simulating the process of core–mantle differentiation, Fe metal alloys and quenched silicate melt have extremely different visual properties and are often easily separated by gently crushing the run product, followed by hand picking separates with the aid of a binocular microscope. The metal phase is typically very easy to mechanically separate, whereas hand picking a pure quenched silicate melt phase is more difficult. The portions of sample that appear to be pure are then placed under a strong magnet and screened thoroughly to ensure that the silicate portions are not at all magnetic. Only the silicate portions that make it through that test are dissolved in strong acids for isotopic analyses. After acid-digestion, the element of interest is chemically separated and purified by column chemistry for isotopic analyses, usually by MC–ICP–MS. Chemical purification procedures will vary for different isotopic systems and phases of interest, as will the analytical instrumentation and measurement procedures. Reviewing these procedures is beyond the scope of this chapter; however excellent reviews for isotopic systems of interest are available in the following chapters of this volume. These post-experiment steps (phase separation and dissolution) are not necessary if the isotopic analyses are conducted in situ by LA-MC–ICP–MS. In this case, the second portion of the sample capsule will be mounted in epoxy and polished prior to laser ablation. To our knowledge, no study has yet physically separated quenched silicate melt from a mineral phase of interest to measure the isotopic fractionation. Ensuring a “clean” mineral phase may prove difficult for experimental run products of this kind. The crystallization kinetics during the experiment will undoubtedly prove important in growing crystals that are both free of melt inclusions or embayments and large enough to mechanically separate from the quenched melt. In cases such as this, in situ analysis may be the best option to ensure that the measured isotopic values are accurate.

CONCLUSIONS The field of experimental non-traditional stable isotope geochemistry has expanded rapidly in the past decade and is likely to continue to grow as more and more natural fractionations in these isotopic systems are uncovered. Indeed, discoveries of fractionations in geologic and planetary materials thus far have greatly outpaced experimental efforts to quantify fractionation factors capable of explaining such fractionations. The current state of the non-traditional stable isotope field can be understood by considering the difficulties of

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using trace element geochemistry in a time when almost no partition coefficients had been determined. Therefore, there is a huge opportunity for creative and careful experimentalists to push this new field forward at a rapid pace. But in order to understand what these experiments can tell us we need to be cautious with interpreting the results. Not all experiments can be compared and often times should not be compared. Changing one variable in the experiment is enough to cause a difference in the equilibrium isotope fractionation and users of the data should be aware of this before extrapolating such data beyond its useful range. Therefore, the most useful path forward is to adopt the approach taken by the experimental trace element partitioning community, wherein the effects of pressure, temperature, solvent and solute composition, oxygen fugacity, and other potential variables are independently evaluated, and parameterized expressions for isotopic fractionation factors are derived. In this way, the community will move toward the goal of understanding the factors that influence equilibrium isotopic fractionations at the relevant conditions in all systems of geologic interest.

ACKNOWLEDGEMENTS A.S. acknowledges NSF Grants EAR1321858 and EAR1464008. We thank the editors, James Watkins, Nicolas Dauphas and Fang-Zhen Teng, for the opportunity to contribute to this volume. The chapter was greatly improved by thoughtful reviews from Laura Wasylenki and Mathieu Roskosz.

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Polyakov VB (2009) Equilibrium iron isotope fractionation at core–mantle boundary conditions. Science 323:912–914 Polyakov VB, Kharlashina NN (1994) Effect of pressure on equilibrium isotopic fractionation. Geochim Cosmochim Acta 58:4739–4750 Polyakov VB, Mineev SD, Clayton RN, Hu G, Mineev KS (2005) Determination of tin equilibrium isotope fractionation factors from synchrotron radiation experiments. Geochim Cosmochim Acta 69:5531–5536 Polyakov VB, Clayton RN, Horita J, Mineev SD (2007) Equilibrium iron isotope fractionation factors of minerals: Reevaluation from the data of nuclear inelastic resonant X-ray scattering and Mossbauer spectroscopy. Geochim Cosmochim Acta 71:3833–3846 Richter FM, Watson EB, Mendybaev RA, Teng FZ, Janney PE (2008) Magnesium isotope fractionation in silicate melts by chemical and thermal diffusion. Geochim Cosmochim Acta 72:206–220 Roskosz M, Luais B, Watson HC, Toplis MJ, Alexander CMO, Mysen BO (2006) Experimental quantification of the fractionation of Fe isotopes during metal segregation from a silicate melt. Earth Planet Sci Lett 248:851–867 Roskosz M, Sio CKI, Dauphas N, Bi WL, Tissot FLH, Hu MY, Zhao JY, Alp EE (2015) Spinel–olivine–pyroxene equilibrium iron isotopic fractionation and applications to natural peridotites. Geochim Cosmochim Acta 169:184–199 Schauble EA (2004) Applying stable isotope fractionation theory to new systems. Rev Mineral Geochem 55:65–111 Schuessler JA, Schoenberg R, Behrens H, von Blanckenburg F (2007) The experimental calibration of the iron isotope fractionation factor between pyrrhotite and peralkaline rhyolitic melt. Geochim Cosmochim Acta 71:417–433 Shahar A, Young ED, Manning CE (2008) Equilibrium high-temperature Fe isotope fractionation between fayalite and magnetite: An experimental calibration. Earth Planet Sci Lett 268:330–338 Shahar A, Ziegler K, Young ED, Ricolleau A, Schauble EA, Fei YW (2009) Experimentally determined Si isotope fractionation between silicate and Fe metal and implications for Earth’s core formation. Earth Planet Sci Lett 288:228–234 Shahar A, Hillgren VJ, Young ED, Fei YW, Macris CA, Deng LW (2011) High-temperature Si isotope fractionation between iron metal and silicate. Geochim Cosmochim Acta 75:7688–7697 Shahar A, Hillgren VJ, Horan MF, Mesa-Garcia J, Kaufman LA, Mock TD (2015) Sulfur-controlled iron isotope fractionation experiments of core formation in planetary bodies. Geochim Cosmochim Acta 150:253–264 Shahar A, Schauble EA, Caracas R, Gleason AE, Reagan MM, Xiao Y, Shu J, Mao W (2016) Pressure-dependent isotopic composition of iron alloys. Science 352:580–582, DOI: 10.1126/science.aad9945 Skulan JL, Beard BL, Johnson CM (2002) Kinetic and equilibrium Fe isotope fractionation between aqueous Fe(III) and hematite. Geochim Cosmochim Acta 66:2995–3015, Sturhahn W (2000) CONUSS and PHOENIX: Evaluation of nuclear resonant scattering data. Hyperfine Interact 125:149–172 Sturhahn W, Toellner TS, Alp EE, Zhang X, Ando M, Yoda Y, Kikuta S, Seto M, Kimball CW, Dabrowski B (1995) Phonon density-of-states measured by inelastic nuclear resonant scattering. Phys Rev Lett 74:3832–3835 Urey HC (1947) The thermodynamic properties of isotopic substances. J Chem Soc:562–581 Wasylenki LE, Swihart JW, Romaniello SJ (2014) Cadmium isotope fractionation during adsorption to Mn oxyhydroxide at low and high ionic strength. Geochim Cosmochim Acta 140:212–26 Wasylenki LE, Howe HD, Spivak-Birndorf LJ, Bish DL (2015) Ni isotope fractionation during sorption to ferrihydrite: Implications for Ni in banded iron formations. Chem Geol 400:56–64 Welch SA, Beard BL, Johnson CM, Braterman PS (2003) Kinetic and equilibrium Fe isotope fractionation between aqueous Fe(II) and Fe(III). Geochim Cosmochim Acta 67:4231–4250 Williams HM, McCammon CA, Peslier AH, Halliday AN, Teutsch N, Levasseur S, Burg JP (2004) Iron isotope fractionation and the oxygen fugacity of the mantle. Science 304:1656–1659 Williams HM, Wood BJ, Wade J, Frost DJ, Tuff J (2012) Isotopic evidence for internal oxidation of the Earth’s mantle during accretion. Earth Planet Sci Lett 321:54–63 Young ED, Galy A, Nagahara H (2002) Kinetic and equilibrium mass-dependent isotope fractionation laws in nature and their geochemical and cosmochemical significance. Geochim Cosmochim Acta 66:1095–1104 Young ED, Manning CE, Schauble EA, Shahar A, Macris CA, Lazar C, Jordan M (2015) High-temperature equilibrium isotope fractionation of non-traditional stable isotopes: Experiments, theory, and applications. Chem Geol 395:176–195

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Reviews in Mineralogy & Geochemistry Vol. 82 pp. 85-125, 2017 Copyright © Mineralogical Society of America

Kinetic Fractionation of Non-Traditional Stable Isotopes by Diffusion and Crystal Growth Reactions James M. Watkins Department of Geological Sciences University of Oregon Eugene, OR USA [email protected]

Donald J. DePaolo Earth Sciences Division Lawrence Berkeley National Laboratory Berkeley, CA USA and Department of Earth and Planetary Science University of California Berkeley, CA USA [email protected]

E. Bruce Watson Department of Geology Rensselaer Polytechnic Institute Troy, NY USA [email protected]

INTRODUCTION Natural variations in the isotopic composition of some 50 chemical elements are now being used in geochemistry for studying transport processes, estimating temperature, reconstructing ocean chemistry, identifying biological signatures, and classifying planets and meteorites. Within the past decade, there has been growing interest in measuring isotopic variations in a wider variety of elements, and improved techniques make it possible to measure very small effects. Many of the observations have raised questions concerning when and where the attainment of equilibrium is a valid assumption. In situations where the distribution of isotopes within and among phases is not representative of the equilibrium distribution, the isotopic compositions can be used to access information on mechanisms of chemical reactions and rates of geological processes. In a general sense, the fractionation of stable isotopes between any two phases, or between any two compounds within a phase, can be ascribed to some combination of the mass dependence of thermodynamic (equilibrium) partition coefficients, the mass dependence of diffusion coefficients, and the mass dependence of reaction rate constants. 1529-6466/17/0082-0004$10.00

http://dx.doi.org/10.2138/rmg.2017.82.4

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Many documentations of kinetic isotope effects (KIEs), and their practical applications, are described in this volume and are therefore not reviewed here. Instead, the focus of this chapter is on the measurement and interpretation of mass dependent diffusivities and reactivities, and how these parameters are implemented in models of crystal growth within a fluid phase. There are, of course, processes aside from crystal growth that give rise to KIEs among non-traditional isotopes, such as evaporation (Young et al. 2002; Knight et al. 2009; Richter et al. 2009a), vapor exsolution (Aubaud et al. 2004), thermal diffusion (Richter et al. 2009a, 2014b; Huang et al. 2010; Dominguez et al. 2011), mineral dissolution (e.g., Brantley et al. 2004; Wall et al. 2011; Pearce et al. 2012; Druhan et al. 2015), and various biological processes (e.g., Zhu et al. 2002; Weiss et al. 2008; Nielsen et al. 2012; Robinson et al. 2014). Coverage of these topics, and how they give rise to KIEs, can be found throughout this volume and in the recent literature.

Organization of the article In the first part of this review, we provide a compilation of the mass dependence of diffusion coefficients in low-temperature aqueous solutions, high-temperature silicate melts, solid metals and silicate minerals. The reader will appreciate both the complexity of isotope diffusion in condensed media as well as the simplicity of the systematic relationships that have emerged, which allow for general predictions regarding the sign and magnitude of isotope fractionation by diffusion in solids and liquids. The second part of this review covers isotope fractionation during crystal growth. We start with models that involve isotope mass dependent diffusion of impurities to a growing crystal. The impurities could be compatible or incompatible elements, provided that they do not affect the growth rate of the crystal itself. We then discuss kinetic isotope fractionation of the stoichiometric constituents of a mineral (e.g., Ca isotopes in CaCO3) due to diffusion as well as surface reaction controlled kinetics, followed by consideration of isotope fractionation of impurities that affect growth rate itself. The presentation includes discussion of three different types of “surface entrapment models,” the underlying mechanisms of mass-dependent reaction rates, and whether isotope fractionation occurs on the aqueous side or the mineral side of the solid–liquid interface. Throughout the chapter, we rely heavily on trace element and stable isotope data for the mineral calcite because of our own familiarity with this mineral and because it is perhaps the best studied phase in the KIE and crystal growth contexts. We note at the outset that while many of the principles developed herein can be transferred to other minerals with similar (desolvation rate-limited) surface reaction mechanisms or growth pathways, additional work is required to adequately describe KIEs for crystals precipitated via non-classical, particle mediated pathways, as described at the end of this chapter. Along the way, it will be seen that molecular dynamics simulations are playing a key role in drawing connections between nanoscale processes and macro-scale observables related to KIEs.

ISOTOPE FRACTIONATION BY DIFFUSION The recognition that diffusion is capable of generating measurable (sub-‰) to large (tens of ‰) isotopic fractionations has catalyzed efforts over the past decade towards figuring out how, when and where diffusion is responsible for isotopic variations in nature (e.g., Ellis et al. 2004; Lundstrom et al. 2005; Beck et al. 2006; Roskosz et al. 2006; Teng et al. 2006, 2011; Dauphas 2007; Jeffcoate et al. 2007; Marschall et al. 2007; Parkinson et al. 2007; Rudnick and Ionov 2007; Bourg and Sposito 2008; Gallagher and Elliott 2009; Dauphas et al. 2010; Sio et al. 2013; Müller et al. 2014; Richter et al. 2014a, 2016; Oeser et al. 2015). In this section, we focus on the progress towards a predictive theory for the mass dependence of diffusion coefficients in aqueous solutions, silicate melts, silicate minerals, and metallic alloys at high temperature.

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Expressions for diffusive fluxes Fick’s first law states that the flux of a chemical species i is directly proportional to the concentration gradient: (1)

Ji = − Di∇Ci , −2 −1

2 −1

where Ji is the flux (moles m  s ), Di is the diffusion coefficient (m  s ), and Ci is the concentration (moles m−3). Note that concentration can be expressed as Ci = rwi / Mi, where ρ is the density of the liquid, wi is the weight fraction of i, and Mi is the molecular weight of i. If the density of the liquid is constant, then concentration gradients can equivalently be expressed in units of wt%. Equation (1) applies to diffusion of a solute in dilute aqueous solution or diffusion of a trace species in an otherwise homogeneous silicate melt. In concentrated solutions or in cases where the diffusing species are unknown, it is customary to define a basis set of chemical components and to recognize that the flux of a component can be driven by concentration gradients in any of the other components. A more general form of Fick’s first law is (Onsager 1945; De Groot and Mazur 1963): n −1

Ji = −∑Dij ∇C j ,

(2)

j =1

where Dij is a matrix of diffusion coefficients. If diffusion of each component i is independent of all other n components, then Dij is a diagonal matrix and each component obeys Equation (1). Generally, this is not the case and the off-diagonal elements of Dij specify the extent of diffusive coupling between the chosen components. The diffusion coefficients Di or Dij are where most of the complication arises in problems involving chemical diffusion. The diffusivity of an element or species depends on the physical properties of the diffusing medium; it may vary spatially in an anisotropic material and it may depend on variables such as temperature, pressure and chemical composition.

Isotopic mass dependence of diffusion in “simple” systems The kinetic theory of gases gives the diffusivity of molecules in an ideal gas as:

1 (3) λn, 3 where v is the mean molecular velocity of particles and λ is the mean free path between collisions. Inserting expressions for v and λ, the tracer diffusion coefficient for a molecular species in a dilute gas is (see Lasaga 1998): D=

D=

8 RT , πM

RT 3 2 π PNd 2

(4)

where T, P, N, d, and M are temperature, pressure, Avogadro’s number, the molecular diameter, and the molecular weight of the gas. This expression is the basis for the square-root-of-mass law, which states that the ratio of diffusion coefficients of two gaseous species is proportional to the inverse square root of their mass; i.e., 1/ 2

D2  m1  =  , D1  m2 

(5)

which is only valid for systems in which the assumptions of kinetic theory (point masses, low pressure such that collisions are infrequent and intermolecular forces are negligible) are approximately valid.

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In condensed systems the effect of mass on diffusion coefficients is considerably more complicated, primarily because the diffusing species have non-negligible potential interactions with their nearest neighbors. Intermolecular potentials are theoretically complex because they depend on the shape and rotation of molecules whose identities are often unknown or are not well defined in systems such as aqueous solutions and silicate melts, as discussed in Watkins et al. (2009, 2011). It has therefore become customary to express the ratio of diffusion coefficients of solute isotopes (D2 / D1) as an inverse power-law function of the ratio of their masses, m2 and m1 (Richter et al. 1999): β

D2  m1  =  , D1  m2 

(6)

where β is a dimensionless empirical parameter. In this review, m will refer to isotopic mass and not the mass of isotopically substituted molecules such as, for example, CO2 or CH4. The β factor is a convenient means of reporting the ratio of isotopic diffusion coefficients because it allows for direct comparison between different elements that have different fractional mass differences between the isotopes (e.g., 7 Li / 6 Li, which differ in mass by about 14%, versus 29Si / 28Si, which differ in mass by about 3%).

Isotopic mass dependence of diffusion in aqueous solution A compilation of β factors for diffusion in aqueous solution is provided in Table 1 and Figure 1. The first takeaway is that isotope fractionation by diffusion in aqueous solution is not nearly as efficient as isotope fractionation by diffusion in a dilute gas, as all but two of the β factors (one measurement for He and one for Ar; see Table 1) are significantly less than the kinetic theory value of 0.5. The second takeaway is that the noble gas elements have larger β factors than the rest of the solutes and their β factors correlate with atomic mass (or radius) such that lighter noble gas elements exhibit greater mass discrimination by diffusion. The low bs for charged species relative to uncharged species, and the dependence of β on atomic size, can be rationalized by considering solute–solvent interactions and the plausible physical mechanisms of diffusion in aqueous solution. The horizontal axis of Figure 1a is the diffusivity of the species normalized by that of H2O. Most of the ionic species diffuse more

0.20

A Ne

0.15

β

K Na Cl

0.05

0.0

Mo Fe CO2-3

Br Mo HCO-3

0.10

100

0.05

CO2

Di / D H 2 O

He Ne

Na

Cs

Li Mg

β

Xe

Zn Ca

B

0.15

Ar

0.10

0.00 10−1

0.20

He

Li

Cl

Xe

Mg

101

0.00

Ar

K

Cs

0.0 Ca

0.2

0.4

0.6

kwex or 1/τ (ps-1)

0.8

1.0

Figure 1. The efficiency of isotope fractionation by diffusion (β) versus metrics for the strength of solute– solvent interactions in aqueous solutions. (a) The β factors correlate with the solvent-normalized diffusivity. Most of the charged species diffuse more slowly than H2O because they interact strongly with their surrounding H2O molecules. (b) Molecular dynamics simulation show that the β factors also correlate with the water exchange rate, kwex, which is equivalent to 1/τ, where τ is the residence time of water molecules in the first hydration sphere surrounding the cation (Bourg et al. 2010). See Table 1 for references.

Table 1. Isotopic mass dependence of diffusion in aqueous solutions T

Description

Isotopic Dsolute system (10−11 m2/s)

Dsolvent (10−11 m2/s) b Noble gases 230 ± 10 0.171 ± 0.028 230 ± 10 0.492 ± 0.122 230 ± 10 0.150 ± 0.018 204 ± 10 0.108 to 0.145 – 0.104 ± 0.031 230 ± 10 0.078 ± 0.029 – 0.055 to 0.074 N/A 0.508 ± 0.036 230 ± 10 0.059 ± 0.023

References

25 °C 25 °C 25 °C 20 °C Room? 25 °C 20 °C Room? 25 °C

MD simulation Experiment MD simulation Experiment Experiment MD simulation Experiment Experiment MD simulation

He He Ne Ne Ne Ar Ar Ar Xe

785 ± 54 722 ± 27 478 ± 37 – – 257 ± 15 – N/A 157 ± 11

75 °C 75 °C 75 °C 75 °C 75 °C 75 °C 25 °C 75 °C 75 °C 75 °C 75 °C 75 °C 21 °C 75 °C 21 °C 20 °C 20 °C

Experiment MD simulation Experiment MD simulation Experiment MD simulation Experiment MD simulation Experiment MD simulation MD simulation Experiment Experiment MD simulation Experiment Experiment Experiment

Ca2+ Ca2+ Mg2+ Mg2+ Li+ Li+ Na+ Na+ K+ K+ Cs+ Cl− ClCl− Br− Fe Zn

– 150 ± 3 – 121 ± 6 – 212 ± 8 – 269 ± 9 – 385 ± 17 404 ± 20 – 114 ± 14 327 ± 14 153 ± 17 57.8 ± 2.3 47.1 ± 0.7

Mo

52 ± 10

204 ± 10

0.0000 ± 0.0010 Malinovsky et al. (2007)

Mo

124 ± 2

204 ± 10

0.0058 ± 0.0019 Malinovsky et al. (2007)

191 ± 7

230 ± 10

0.039 ± 0.009

Jähne et al. (1987)

204 ± 35

230 ± 10

0.01 to 0.14

Zeebe (2011)

110 ± 18

230 ± 10

−0.04 to 0.17

Zeebe (2011)

80 ± 18

230 ± 10

−0.04 to 0.13

Zeebe (2011)

N/A

204 ± 10

0.067 to 0.090

Knox et al. (1992)

N/A

204 ± 10

0.020 to 0.026

Fuex (1980)

N/A N/A

204 ± 10 204 ± 10

0.069 to 0.093 0.050 to 0.066

Benson and Krause (1980) Benson and Krause (1980)

Diffusion of 20 °C, Mo7O246− and pH ~0.5 Mo8O264− 20 °C, Diffusion of pH ~7 MoO42− 25 °C

Experiment

25 °C

MD simulation

25 °C

MD simulation

25 °C

MD simulation

20 °C

Gas–water exchange exp.

20 °C

Experiment

20 °C 20 °C

Experiment Experiment

CO2 (13C) CO2 (13C) HCO3− (13C) CO32− (13C) H2 CH4 (13C) O2 N2

Ionic species 590 ± 10 0.0045 ± 0.0005 590 ± 10 0.0000 ± 0.0108 590 ± 10 0.0000 ± 0.0015 590 ± 10 0.006 ± 0.018 590 ± 10 0.015 ± 0.002 590 ± 10 0.0171 ± 0.0159 – 0.023 ± 0.023 590 ± 10 0.029 ± 0.0.022 590 ± 10 0.042 ± 0.002 590 ± 10 0.049 ± 0.017 590 ± 10 0.030 ± 0.018 590 ± 10 0.0258 ± 0.0144 209 ± 10 0.0296 ± 0.0027 590 ± 10 0.034 ± 0.018 209 ± 10 0.025 ± 0.005 204 ± 10 0.0025 ± 0.0003 204 ± 10 0.0019 ± 0.0003 Molecular species

Bourg and Sposito (2008) Jahne et al. (1987) Bourg and Sposito (2008) Tempest and Emerson (2013) Tyroller et al. (2014) Bourg and Sposito (2008) Tempest and Emerson (2013) Tyroller et al. (2014) Bourg and Sposito (2008) Richter et al. (2006) Bourg et al. (2010) Richter et al. (2006) Bourg and Sposito (2007) Richter et al. (2006) Bourg and Sposito (2007) Pikal (1972); Richter et al. (2006) Bourg et al. (2010) Bourg et al. (2010) Bourg et al. (2010) Bourg et al. (2007) Richter et al. (2006) Eggenkamp and Coleman (2009) Bourg and Sposito (2007) Eggenkamp and Coleman (2009) Rodushkin et al. (2004) Rodushkin et al. (2004)

Notes 1. Water diffusivity comes from Bourg and Sposito (2007) (their Fig. 1) 2. For gas-water exchange experiments, the range in b represents the range of n = 0.5 to 0.67, where n is related to the dynamics of the air-water interface (cf. Tempest and Emerson 2013)

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slowly than H2O (Di / DH2O 100 ºC) associated with volcanic lakes and/or hydrothermal systems (Eggenkamp 1994; Barnes et al. 2009a; Sharp et al. 2010a). Theoretical calculations show that 1000 ln αHCl–Cl(aq) is less than 2‰ at all temperatures (Schauble et al. 2003), failing to explain these high values. Fractionation between HCl(g) and Cl−(aq) has been determined experimentally by measuring the isotopic composition of HCl vapor equilibrated with 1 M HCl solution and confirm these theoretical calculations (Fig. 1, Sharp et al. 2010a). However, this experiment evidently does not mimic the dynamic system of a volcanic fumarole. Flowthrough experiments, in which air is bubbled through hydrochloric acid and the removed HCl vapor is allowed to condense along the length of a cool tube, produced fractionations as large as +10‰. These high values are due to kinetic exchange between the solvated Cl− ion in the aqueous solution condensing along the walls of the tube and the flowing 37Cl-enriched HCl vapor (Sharp et al. 2010a). In contrast to preferential loss of 37Cl into HCl, 35Cl has been shown to be lost to the vapor phase during HCl evaporation experiments due to its higher translation velocity (greater escaping tendency) and vapor pressure compared to 37Cl (Sharp et al. 2010a). Chlorine isotopes can be used as an indicator of volcanic processes such as degassing, but care must be taken to recognize the possibility of Cl loss due to HCl degassing when using Cl isotopes as a tracer of source (Rizzo et al. 2013; Barnes et al. 2014b; Fischer et al. 2015). The largest kinetic isotope effects measured to date are seen in lunar samples. d37Cl values as high as 34‰ have been measured in lunar basalts (Sharp et al. 2010b; Tartèse et al. 2014; Treiman et al. 2014), and are thought to be a result of rapid diffusion of Cl-species to the vapor phase during degassing (see section on the Moon for details).

CHLORINE ISOTOPIC COMPOSITION OF VARIOUS GEOLOGIC RESERVOIRS In general, the chlorine isotope variation in nature is relatively small, ranging from ~ −2 to +2‰, due to small equilibrium fractionation factors. However, large variations are observed, e.g., in extraterrestrial materials and volcanic gases, due to kinetic fractionation (Fig. 3). Below we summarize the variations in δ37Cl values observed in natural materials. The isotopic composition of many of these Cl reservoirs has only been recently determined and, in some cases, is not universally agreed upon. All δ37Cl values discussed below were determined by IRMS, unless indicated as pTIMS or SIMS measurements.

Mantle/OIB/mantle derived material The first systematic Cl isotope analyses of Mid-Ocean Ridge Basalts (MORB) averaged −0.2 ± 0.5‰ (n = 12) and showed no variation with Cl content (Sharp et al. 2007). Analysis of sub-continental mantle samples were indistinguishable (−0.03 ± 0.25‰). These results supported previous analyses of mantle-derived carbonatites with d37Cl values averaging −0.2 ± 0.4‰ (Eggenkamp and Koster van Groos 1997) and are similar to seawater and evaporites. The following year, Bonifacie et al. (2008a) analyzed MORB samples and found a positive correlation between the d37Cl value and Cl content. They concluded that the mantle has a d37Cl value

Barnes & Sharp

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low-T

high-T

interact with seawater

interact with seds

-15

-10

-5

0

5

seawater porewaters brines/formation waters natural perchlorates evaporites sediments/sedimentary rocks metasedimentary rocks altered oceanic crust serpentinites obducted serpentinites volcanic ashes/lavas volcanic gases MORB OIB chondritic meteorites Moon Mars

10 15 20 δ37Cl (‰; vs SMOC)

25

30

35

Figure 3. Chlorine isotope variability in different terrestrial and extraterrestrial reservoirs. Dashed line is seawater (SMOC). Wider gray band delineates −0.5 to +0.5‰. Almost all evaporite and MORB values fall within this gray band illustrating that the largest Cl terrestrial reservoirs are near 0‰. Triangles are SIMS analyses; circles are IRMS analyses. Each symbol represents one sample, with the exception of some SIMS analyses which are individual analyses of different mineral grains within the same sample. For multiple analyses of the same sample, only the average is shown. All IRMS data are bulk δ37Cl values, unless indicated otherwise. Sediments/sedimentary rocks: gray circles are marine; open circles are non-marine. Almost all marine sediments are negative; non-marine are negative and positive. OIB: gray triangles = EM1; open triangles = EM2; black triangles = HIMU. Chondritic meteorites: gray circles = ordinary chondrites; open circles = carbonaceous chondrites; black circles = enstatite chondrites. Moon = gray circles = water-soluble chloride; open circles = structurally bound chloride. Data sources: porewaters: Ransom et al. (1995), Hesse et al. (2000), Godon et al. (2004), Bonifacie et al. (2007); brines/formation waters: Kaufmann et al. (1987, 1988, 1993), Eggenkamp (1994), Eastoe et al. (1999, 2001), Zeigler et al. (2001), Shouakar-Stash et al. (2007), Zhang et al. (2007), Stotler et al. (2010); natural perchlorates: Böhlke et al. (2005); Sturchio et al. (2006); Jackson et al. (2010); evaporites: Eastoe et al. (1999, 2001, 2007), Eastoe and Peryt (1999), Eggenkamp et al. (1995), Arcuri and Brimhall (2003); sediments/sedimentary rocks: Acuri and Brimhall (2003), Barnes et al. (2008; 2009), Selverstone and Sharp (2015); metasedimentary rocks: John et al. (2010), Selverstone and Sharp (2013, 2015), Barnes et al. (2014); altered oceanic crust: serpentinites: Barnes and Sharp (2006), Bonifacie et al. (2007, 2008), Barnes et al. (2008, 2009), Barnes and Cisneros (2012), Boschi et al. (2013); obducted serpentinites: Barnes et al. (2006, 2013, 2014), Bonifacie et al. (2008), John et al. (2011), Selverstone and Sharp (2013); volcanic ashes/lavas: Barnes et al. (2008, 2009), Barnes and Straub (2010), Chiaradia et al. (2014), Cullen et al., (2015), Rizzo et al. (2013); volcanic gases: Barnes et al. (2008, 2009), Rizzo et al. (2013), MORB: Sharp et al. (2007); OIB: John et al. (2010); chondritic meteorites: Sharp et al. (2007, 2012); Moon: Sharp et al. (2010), Tartese et al. (2014), Treiman et al. (2014), Boyce et al. (2015); Mars: Sharp et al. (2016).

of ≤ −1.6‰, arguing that the higher d37Cl values represent seawater contamination. This trend was not observed in the earlier study. Analyses of additional mantle/mantle-derived material (e.g., alkali inclusions in fibrous diamonds) as well as an extensive suite of chondrites were all close to 0‰; the mantle-derived samples were −0.2 ± 0.3‰, analytically indistinguishable from the chondrite value of −0.3 ± 0.3‰ (Sharp et al. 2013c). The authors suggested that the low δ37Cl values analyzed previously may not reflect uncontaminated samples, but rather were an analytical artifact associated with analysis of small samples in dual inlet mode.

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Mantle Cl isotope heterogeneities are not in doubt. The mantle likely preserves some heterogeneities due to the subduction of crustal material. Using SIMS analyses of Ocean Island Basalt (OIB) glasses, John et al. (2010) showed that δ37Cl values range from −1.6 to +1.1‰ for HIMU-type and −0.4 to +2.9‰ for EM-type lavas, and positively correlate with 87Sr / 86Sr ratios. Sharp et al. (2007) report values for two MORB glasses from a similar location in the East Pacific Rise that are distinctly lower (−1‰) than the near zero values of other MORB samples. They may also represent a contaminated mantle origin.

Seawater and seawater-derived chloride The oceans and associated ocean-derived Cl in the form of evaporates, sedimentary pore fluids, and marine aerosols are the second largest Cl reservoir on Earth (e.g., Schilling et al. 1978; Jarrard 2003; Sharp and Barnes 2004). As discussed in the Chlorine Isotope Nomenclature and Standards section, the Cl isotope composition of seawater is homogeneous (Godon et al. 2004b) due to its extraordinarily long residence time in the ocean, with a defined d37Cl value of 0‰ (Kaufmann et al. 1984). The chlorine isotope composition of the ocean has remained relatively constant through time based on the isotopic composition of Phanerozoic halite and Precambrian cherts, barite, and dolomite (Eastoe et al. 2007; Sharp et al. 2007; Sharp et al. 2013c) (Fig. 4). First-formed halite in equilibrium with seawater at 22 ± 2 °C will have a δ37Cl value of +0.26 ± 0.07‰ (Eggenkamp et al. 1995). If seawater is isolated and allowed to completely evaporate, the d37Cl value of the residual salt will have a bulk d37Cl value identical to seawater. As salt is precipitated from the fluid, the fluid itself becomes progressively enriched in 35Cl. If this fluid is then physically removed from the salt, any redeposition of salt from the 35Cl-enriched fluid will also produce a 35Cl-enriched salt. Naturally occurring halite formed from seawater via equilibrium fractionation is reported to have δ37Cl values ranging from −0.6 to +0.4‰. This range of values is due to multiple precipitation and dissolution events of the evaporite minerals. δ37Cl values of halite outside of this range require an alternate explanation, typically attributed to the influx of a non-zero ‰ fluid (Eastoe et al. 1999; Eastoe and Peryt 1999). Sedimentary pore fluids have some of the lowest reported δ37Cl values, ranging from −7.8 to +0.3‰ with the vast majority of the samples being negative (Ransom et al. 1995; Hesse et al. 2000; Spivack et al. 2002; Godon et al. 2004a; Bonifacie et al. 2007a). Multiple explanations have been proposed for the predominantly negative values, including mineral-fluid interaction, ion filtration, diffusion, and melting of gas hydrates (Ransom et al. 1995; Hesse et al. 2000; Spivack et al. 2002; Godon et al. 2004a; Bonifacie et al. 2007a). Although these processes may, in part, contribute to the low d37Cl values of pore waters, none of these mechanisms should lower the d37Cl values of seawater-derived fluids to such a large extent. Authigenic clay minerals do not generally have positive d37Cl values (Barnes et al. 2008; Barnes et al. 2009a; Selverstone and Sharp 2015), so their formation could not produce a negative residual fluid. No diffusion studies have documented Cl-isotope changes in natural systems greater than ~3‰. Ion filtration has been invoked to explain values of natural samples as low as ~ −5‰ (Ziegler et al. 2001; Godon et al. 2004a). However, in most cases, the negative extreme remains unexplained. An alternative explanation for the low d37Cl values may be related to kinetic fractionation associated with the formation of organochlorine compounds. Up to 20% of marine ‘natural products’ are organohalogens (Gribble 2010). Organohalide microbial communities may play an important role in the carbon and chlorine cycle in marine sediments and have been shown to be extensively distributed in sediments of the Nankai Trough subduction zone (Futagami et al. 2013). High organochlorine concentrations have also been found in sediment traps from the Arabian Penninsula (Leri et al. 2015). Organic matter incorporates Cl in the +1 oxidation state, as opposed to dissolved Cl− in the −1 oxidation state. As such, it should strongly incorporate 37 Cl relative to 35Cl. However, in natural systems kinetic fractionation often results in an enrichment of 35Cl in organic matter in excess of 10‰ (Reddy et al. 2002; Aeppli et al. 2013). Marine organic-rich sediments may therefore have strong negative d37Cl values. As they

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are degraded (microbially or through inorganic oxidation), they could release Cl into the porewaters with a very low d37Cl value. Organic materials may be responsible for the low d37Cl values of marine sedimentary pore fluids and may be the prime driver in the creation of low d37Cl values on Earth. The role of organic material is mostly unexplored and may be critical to Cl fractionation in sedimentary systems. Marine aerosols form from the interaction of sea-spray NaCl with atmospheric sulfuric and nitric acids. Cl− in the aerosols are replaced by sulfate or nitrate ions, releasing HCl in the process. Marine aerosols have δ37Cl values, determined by pTIMS, ranging from −0.85 to +2.53‰ (Volpe and Spivack 1994; Volpe et al. 1998). In general, δ37Cl values of the aerosols increase as chlorine loss increases (Volpe and Spivack 1994). These values are interpreted to be due to fractionation during volatilization of HCl (Volpe and Spivack 1994); however, the fractionation due to HCl exchange has been shown to have the opposite sign (Schauble et al. 2003; Sharp et al. 2010a). Subsequent works show that smaller particles have positive values and larger particles have negative values, suggesting kinetic effects likely play a role in release of Cl from marine aerosols (Volpe et al. 1998).

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Sediments Our knowledge of the Cl isotope composition of sediments is limited to only a handful of bulk analyses. Marine sediments from the Pacific Ocean and sampled by Ocean Drilling Program have δ37Cl values ranging from −2.5 to +0.7‰ with the majority of the samples being negative (average = −0.8 ± 0.8‰; n = 24) (Barnes et al. 2008; Barnes et al. 2009a). Jurassic sedimentary rocks deposited in a shallow marine basin from Chile have also have primarily negative δ37Cl values (−2.6 to +0.5‰; average = −0.9 ± 0.9‰; n = 17) (Arcuri and Brimhall 2003). Marine shales from the Swiss Alps are isotopically negative with bulk δ37Cl values of −3.0 to −0.7‰; whereas, interlayered playa-facies sedimentary rocks range from 0 to +1.8‰ and fluvial/deltaic facies rocks have δ37Cl values between −2 and −1‰ (Selverstone and Sharp 2015). Intra-bed samples are generally isotopically homogeneous, however, great isotopic variably occurs between sedimentary beds. Isotopic differences are interpreted to be due to variations in protolith composition. Protolith variability may reflect biological processes, evaporation of source waters, and input of an external Cl source (e.g., aerosols, formation waters) (Selverstone and Sharp 2015). In sum, there is no unambiguous chlorine isotopic “signature” for sedimentary material; however, all organic-rich marine sediments and their lithified equivalents that have been measured to date have negative δ37Cl values. Non-marine sedimentary material can have either positive or negative δ37Cl values. In order to further investigate isotopic variability and possible fractionation during subduction, a few studies have analyzed metasedimentary rocks. High-pressure (HP) marine metasedimentary rocks from Ecuador which have δ37Cl values ranging from −2.2 to +2.2‰ (average = 0.0 ± 1.7‰; n = 8), leading the authors to speculate that the high δ37Cl values may be due to preferential loss of 35Cl during prograde metamorphism (John et al. 2010). However, no such high values are reported in a suite of HP and ultra-high-pressure (UHP) metasedimentary rocks of marine origin from the Western Alps—δ37Cl values range from −3.6 to +0.0‰ (average = −2.3 ± 0.9‰; n = 25), overlapping with modern marine sediments and extending to lower values (Selverstone and Sharp 2013). These low values are interpreted to have been derived from interaction with isotopically negative sedimentary pore fluids expelled from the accretionary wedge during plate bending and subduction, rather than isotopic fractionation during subduction (Selverstone and Sharp 2013). In order to address Cl isotope fractionation over a range of metamorphic temperatures, Selverstone and Sharp (2015) determined the Cl isotope composition of unmetamorphosed marine and non-marine sedimentary rocks (see above) and their metamorphosed equivalents through increasing metamorphic grade. The metamorphosed equivalents overlap with the respective protoliths supporting minimal modification of the Cl isotope composition during Alpine metamorphism, despite significant Cl and H2O loss (Selverstone and Sharp 2015). HP and UHP metamorphosed sedimentary rocks retain their original isotopic composition despite devolatilization unless infiltrated by an externally derived fluid. Future work, particularly on the role of organochlorides, is necessary to fully understand the isotopic variability of sedimentary materials.

Altered Oceanic Crust (AOC) Hydrothermal alteration of oceanic crust results in the formation of secondary hydrous minerals (e.g., clays, chlorite, amphiboles), many of which can host high concentrations of Cl (amphibole is the dominant Cl host containing up to 4 wt. % Cl) (e.g., Ito et al. 1983; Vanko 1986; Philippot et al. 1998). A detailed study by Barnes and Cisneros (2012) report δ37Cl values and concentrations for 50 altered basalt (extrusive lavas and sheeted dikes) and gabbro samples from seven DSDP/ODP/IODP sites. δ37Cl values range from −1.4 to +1.8‰. In general, chlorine concentrations and δ37Cl values increase with increasing temperature of alteration and amphibole content. Positive δ37Cl values in amphibole-rich samples are consistent with theoretical calculations and preliminary experimental data that indicate amphibole should be

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slightly enriched in 37Cl compared to co-existing fluid (Schauble et al. 2003; Cisneros 2013). Nevertheless, the very high δ37Cl values cannot be explained solely by equilibrium fractionation at high temperatures. The samples with negative δ37Cl values are dominated by clay minerals (Fig. 5). The correlation between chlorine isotope composition/chlorine concentration and modal mineralogy suggests that chlorine chemistry is a rough indicator of metamorphic grade in altered oceanic crust (Barnes and Cisneros, 2012). Bonifacie et al. (2007b) also analyzed AOC samples from ODP Hole 504B which had δ37Cl values ranging from −1.6 to −0.9‰ (n = 3). These values are all lower than the δ37Cl values from Hole 504B (n = 9) reported by Barnes and Cisneros (2012), although none of the analyses are on the exact same sample. Until inter-laboratory calibrations are more universal, it is difficult to discuss analytical differences between labs.

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Figure 5. Variation in chlorine geochemistry and mineralogy in Ocean Drilling Program Hole 735B. A) Variation of Cl concentration (wt.%) and B) Cl isotope composition with depth. C) Variation in the modal abundance of amphibole and smectite as bulk volume percent with depth (Dick et al. 2000) and volume percentage of amphibole and chlorite on bulk sample XRD powders (Barnes and Cisneros 2012). D) Alteration index determined by the percentage of amphibole replacing pyroxene (Stakes et al. 1991). E) Volume percent abundance of higher temperature amphibole veins and lower temperature veins (clays, carbonates, zeolites, prehnite) (Bach et al. 2001). Figure reprinted with minor alterations from Chemical Geology Vol. 326−327 Barnes JD and Cisneros M. Mineralogical control on the chlorine isotope composition of altered oceanic crust, p. 51−60 (2012) with permission from Elsevier.

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Serpentinites The chlorine isotope composition of serpentinites (n > 200) has been better characterized than most of the above discussed reservoirs. The chlorine isotope composition of seafloor serpentinites (sampled by the DSDP/ODP/IODP), including serpentinite clasts extruded at serpentine seamounts, typically have δ37Cl values ranging from ~ −1.6 to +0.5‰ (Barnes and Sharp 2006; Barnes et al. 2008, 2009b; Bonifacie et al. 2008b; Boschi et al. 2013). Cl is hosted as both a water-soluble phase (thought to be halite precipitated along grain boundaries) and structurally bound within the serpentine structure. Structurally bound Cl is ~ +0.2‰ higher than water-soluble Cl (Sharp and Barnes 2004; Barnes and Sharp 2006), consistent with predicted 37Cl enrichment in silicate hydrous phases relative to salts (Eggenkamp et al. 1995; Schauble et al. 2003). Seafloor serpentinites can be divided into two populations based on tectonic setting and fluid source. The vast majority of analyzed seafloor serpentinities are isotopically positive due to hydration via seawater along faults and fissures, in which the serpentine preferentially incorporates 37Cl into its structure (Barnes and Sharp 2006). Isotopically negative serpentinites are interpreted to have formed via hydration from isotopically negative pore fluids. These serpentinites are typically found juxtaposed to sediments via thrust faults. The isotopically negative pore fluids hydrate the peridotite resulting in an isotopically negative serpentinite (Barnes and Sharp 2006). Obducted serpentinites have δ37Cl values overlapping seafloor serpentinite values and some extending to higher values (−1.5 to +2.4‰) (Barnes et al. 2006; Barnes et al. 2013, 2014a; Bonifacie et al. 2008b; John et al. 2011; Selverstone and Sharp 2011, 2013). Despite significant volatile loss, original seafloor serpentinite δ37Cl values are preserved throughout prograde metamorphism (Barnes et al. 2006; Bonifacie et al. 2008b; John et al. 2011; Selverstone and Sharp 2013), similar to observations in metasedimentary rock sequences (Selverstone and Sharp 2015). However, Cl isotope compositions can be modified from original values due to infiltration of an externally derived fluid, as exemplified by the high δ37Cl values of some obducted serpentinites. The sources responsible for these high δ37Cl values are not clear, but in some cases are speculated to derive from nearby sediments (Selverstone and Sharp 2011, 2013; Barnes et al. 2014a).

Perchlorates Perchlorates are an oxidized form of Cl that form by photochemical reactions in the stratosphere (Bao and Gu 2004). The equilibrium fractionation between chloride (Cl−) and perchlorate (ClO4−, oxidation state of +7) is enormous. Harold Urey calculated the ClO4−–Cl− fractionation to be 80‰ at 23 °C (Urey 1947). Schauble et al. (2003) estimated the fractionation to be 70‰ at 25 °C. Hoering and Parker (1961) first measured the d37Cl values of perchlorates and found them to be indistinguishable from the normal range of chlorides. They concluded that perchlorates and chlorides from the same nitrate deposits from Tarpaca, Chile were not in isotopic equilibrium. The formation mechanism of perchlorates was assumed to be one of oxidation of Cl− to ClO4− without isotopic fractionation. Preservation of natural perchlorates on Earth is rare. Because they are soluble in water, they are found primarily in arid environments (Jackson et al. 2015b). Large deposits are known from several notable locations around the world, including the Antarctic dry valleys, deserts in the American southwest (Death Valley, Amargosa Desert), the Atacama Desert (Chile), and China. The isotopic composition of the Atacama perchlorates range from −15 to −9‰ (Sturchio et al. 2006; Böhlke et al. 2009), whereas perchlorates from the Southwest United States have d37Cl values of −3 to +6‰, similar to their presumed chloride source (Sturchio et al. 2012a). Perchlorates from groundwater samples from the Southern High Plains of Texas and New Mexico that are thought to be of natural origin have d37Cl values of 3.1 to 5.1‰ (Jackson et al. 2010).

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Laboratory experiments involving bacterial reduction of perchlorate by Azospira suillum indicates a large kinetic fractionation (Coleman et al. 2003; Sturchio et al. 2003). As is seen in the majority of bacterially mediated reactions, the light isotope is preferentially reduced to Cl−, resulting in increasing d37Cl values of the remaining perchlorate. The 1000 ln ε value (perchloratechloride) value (where ε represents a kinetic fractionation factor given by ε =  Rperchlorate / Rchloride, and R = 37Cl / 35Cl) determined from these studies span a range of −15.8 to −14.8‰ at 37 °C (Coleman et al. 2003) and −16.6 to −12.7‰ at 22 °C (Sturchio et al. 2003). In a more recent study in which both Cl and O isotope ratios were monitored, a 1000 ln ε value of −13.2‰ was obtained, independent of temperature or bacterial strain (Sturchio et al. 2007). The high d37Cl values measured in the Southern High Plains groundwaters are interpreted as the result of bacterial reduction (Jackson et al. 2010). The low d37Cl values of the Atacama samples remains unexplained. Synthetic perchlorates have a narrow range of d37Cl values of −3.1 to +2.3‰ (Ader et al. 2001; Bao and Gu 2004; Böhlke et al. 2005). The similarity to natural salt deposits suggests that the formation of perchlorate by electrolysis of NaCl solutions has a  +1.0‰). Future work is needed to better constrain Cl isotope behavior during fluid-rock interaction and volcanic degassing in order to assess volatile sources to arc magmatism. Not all studies of Cl in subduction zones have focused on volcanic and associated spring outputs. A study of metasomatized, suprasubduction-zone mantle (Finero peridotite, Ivrea Zone, Italy) used Cl isotopes to distinguish two slab-derived fluid infiltration events into the overlying mantle wedge: one, with a d37Cl value ≤ −2‰ and the other with a d37Cl value ≥ +2‰ (Selverstone and Sharp 2011). A subsequent study on exhumed subduction zone rocks of the Zermatt–Saas ophiolite, Italy, documents infiltration of pore fluids during the initial stages of subduction (Selverstone and Sharp 2013).

Crustal fluids Chlorine isotopes have been used to trace sources and hence subsurface flow and transport mechanisms of groundwater, primarily basinal brines and formation waters associated with oil fields. These waters have a wide range of d37Cl values from −4.25 to +1.54‰ (Fig. 10). In many cases these d37Cl values reflect mixing of multiple fluid sources and evaporation/halite dissolution. As discussed in the Equilibrium Cl isotope fractionation section, halite precipitation from seawater produces halite with a d37Cl value of +0.26‰ (Eggenkamp et al. 1995), lowering the d37Cl value of the residual fluid. If evaporation proceeds until the formation of K–Mg chlorides (f = 0.3), which

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are depleted in 37Cl relative to the saturated solution, the d37Cl values of the solution can then begin to increase (Fig. 11A) (Eggenkamp et al. 1995; Eastoe et al. 1999). Multiple evaporative cycles (e.g., dissolution due to influx of water followed by evaporation) can result in d37Cl values > +0.3‰, although little of the original halite remains. Removal of about half of the original halite results in a shift of ~0.15‰ (Fig. 11B) (Eastoe and Peryt 1999; Eastoe et al. 2007). Mixing and dissolution–precipitation processes do not explain the full range of crustal fluid values, which require additional fractionation processes such as diffusion and ion filtration. As discussed in the Kinetic Cl isotope fractionation section, diffusion due to large concentration gradients can result in a shift in several per mil due to the greater translational velocity of light isotopologues. The minimum d37Cl values strongly depends on the initial Cl concentration of the “fresher” fluid (Fig. 11C) (Eggenkamp 1994). Shifts greater than ~3‰ in natural samples cannot be explained solely in terms of diffusion, in part because concentration gradients are typically too small. Ion filtration has been invoked to explain larger shifts in d37Cl values, up to ~ −5‰ (Ziegler et al. 2001; Godon et al. 2004a), in natural samples. An isotopic shift of this magnitude requires a ~20 times increase in Cl concentration behind the membrane (Phillips and Bentley 1987) (Fig. 11D). While not considered in any crustal fluid study, organohalides may partly explain the very negative d37Cl values measured. As discussed in the Isotopic Fractionation section, kinetic effects result in very negative d37Cl values of organic matter (down to −12‰). Degradation of this material would result in fluids with isotopically negative values. Additional work evaluating the importance of organic matter as a means of fractionating Cl isotopes is warranted. Figure 12 summaries the approximate maximum magnitude of change in crustal fluid d37Cl values in response to the above discussed mechanisms. Basinal brines and formation waters from Texas and the Gulf Coast of the United States have d37Cl values of −1.9 to +0.7‰ (Kaufmann et al. 1988; Eastoe et al. 1999, 2001). Three component mixing between 1) seawater; 2) an isotopically positive brine due to halite dissolution; and 3) an isotopically negative brine (due to halite precipitation and/or diffusion) explains the full range of d37Cl values (Kaufmann et al. 1988; Eastoe et al. 1999, 2001).

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Formation waters from the Siberian Platform have overlapping d37Cl values, but extend to more positive values (−0.67 to +1.54‰) (Shouakar-Stash et al. 2007). As with the Gulf Coast waters, negative values are thought to result from evaporation of seawater, with the resulting fluids having negative values due to the precipitation of isotopically positive halite.

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Figure 13. Ca/Cl, K/Cl, and d37Cl value of Michigan Basin formation waters (open and solid squares) (Kaufmann et al. 1993). The formation waters show possible mixing trends (dashed gray line) between the deeper basin waters and Canadian Shield waters (gray circles) (Kaufmann et al. 1987); however additional, and more isotopically negative shield waters, make the mixing argument less robust (small gray circles) (Stotler et al. 2010).

Some positive values suggest dissolution of halite; however, the authors point out that the high values are above any evaporite samples and instead suggest mixing with some additional heavy reservoir. The source of the very positive values is uncertain and may involve fluidrock interaction or permafrost freezing effects (Shouakar-Stash et al. 2007). Formation waters from the Michigan Basin have d37Cl values ranging from −1.2 to +0.1‰ and show nice correlations with Ca/Cl and K/Cl, suggesting mixing between isotopically negative deep basinal brines and isotopically positive Canadian Shield waters (Fig. 13) (Kaufmann et al. 1987, 1993). Diffusional processes have been used to explain isotopically negative (as low as −2.5‰) groundwaters from the Great Artesian Basin in Australia (Zhang et al. 2007); whereas, extreme negative formation waters from the North Sea (as low as −4.25‰; Eggenkamp, 1994) necessitate ion filtration effects (Ziegler et al. 2001) or organochloride contribution.

Tracer in ore deposits Hydrothermal ore fluids may be a mixture of magmatic, metamorphic, meteoric, and groundwater derived fluids. These hybridized fluids transport metal ions responsible for exploitable deposits; therefore, determining fluid source and percent contribution of each source is of economic importance. Early work recognized the potential for Cl isotopes to be used as a tracer of fluid source in ore deposits, yet its application remains relatively under-utilized. The d37Cl variations observed in ore deposits cover a range of over 6‰, with very low values of  90% yield. Mass spectrometry. For TIMS, the measurements are relatively straightforward. Sub-microgram to a few microgram quantities of Cr are loaded onto Re filaments with silica gel and saturated boric acid, with or without Al. Chromium is steadily ionized at ~1200–1400 °C. Within this temperature range, the ionization efficiency of Fe is very low. Although Fe signals can be detected if there is any Fe left over from the sample preparation procedure, they can usually be corrected by monitoring the 56Fe signal. A study showed that interferences from Fe are correctable even when the 56Fe/52Cr signal ratio is as high as 10−4 (Qin et al. 2010a). The other major isobaric interference, Ti, is not ionized in the temperature range where Cr is ionized, generating no signal above background on a Faraday cup, which has a detection limit of 0.02 mV with a 4.2-second integration time. Given the extremely low natural abundance of 50V relative to 51V and removal of the majority of V during column procedures, V is usually not problematic for either TIMS or ICP–MS measurements. Regardless, any potential isobaric interferences from Fe, Ti, and V can be monitored and corrected by measuring 56Fe, 51V, and 49Ti signals. Because selective ionization is not possible, isobaric interferences from Fe and Ti can be much more severe for MC–ICP–MS than for TIMS. Fe, Ti, and V need to be eliminated by chemistry as much as possible before performing mass spectrometry. In addition to direct isobaric interferences from Fe, Ti, and V, molecular (polyatomic) interferences associated with Ar (the typical primary gas in the plasma source) such as 40Ar12C on 52Cr, 40Ar14N on 54 Cr, and 40Ar16O on 56Fe, are also unavoidable. The measurements are usually conducted using desolvating systems to minimize the formation of these molecular interferences. The measurements must be conducted using a high-mass-resolution instrument (resolving power greater than ~6,000) to distinguish the peaks of molecular interferences from those of Cr and Fe (Weyer and Schwieters 2003). MC–ICP–MS instruments built before 2000 could not offer this resolution (see Weyer and Schwieters 2003). At the present time, new-generation TIMS (Triton and Triton plus) have provided the most precise (Table 1) and efficient mass-independent measurements of Cr isotopes (Trinquier et al. 2008a; Qin et al. 2010a). To achieve a comparable level of precision, a several-fold increase in Cr quantity is required for MC–ICP–MS measurements (Schiller et al. 2014). For mass-dependent measurements, assuming a Cr double spike is used, choosing between TIMS and MC–ICP–MS depends on several factors: (1) TIMS can achieve higher precision using less Cr than MC–ICP– MS; therefore, for samples (such as seawater) where only a small amount (nanogram quantities)

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Table 1. Comparison of precision of Cr isotopic measurements from different laboratories. Reference

Instrument

Quantity of Cr used

Reported precision

Double spike

Ball and Bassett (2000)

VG−336 and MAT−261 (TIMS)

665 ng

—

No

Ellis et al. (2002)

VG−354 and MAT−261 (TIMS)

200 ng

±0.2‰

Yes

Schoenberg et al. (2008)

Neptune(MC–ICP–MS)

2 µg

±0.048‰

Yes

Halicz et al. (2008)

Neptune(MC–ICP–MS)

10 µg

±0.06‰

No

Frei et al. (2009)

IsotopX/GV IsoProbe(TIMS)

2−5 µg

±0.08‰

Yes

Bonnand et al. (2011)

Neptune(MC–ICP–MS)

250 ng

±0.059‰

Yes

Han et al. (2012)

Isoprobe(MC–ICP–MS)

1 µg

±0. 15‰

Yes

Farkaš et al. (2013)

Neptune(MC–ICP–MS)

2 µg

±0.066‰

Yes

Schiller et al. (2014)

Neptune(MC–ICP–MS)

30−60 µg

±0.011‰

No

Shen et al. (2015)

Neptune plus(MC–ICP–MS)

1 µg

±0.04‰

Yes

Bonnand et al. (2016)

Neptune(MC–ICP–MS)

2 µg

±0.022‰

Yes

Zhang et al. (2016) (submitted) Neptune plus(MC–ICP–MS)

1 µg

±0.05‰

Yes

Wang et al. (2016) (in revision)

Neptune plus (MC–ICP–MS)

10 ng

±0.26‰

Yes

Scheiderich et al. (2015)

Thermo Triton TIMS

20–600 ng

0.026‰

Yes

of Cr can be extracted, TIMS is a better choice. (2) TIMS instruments are less complex than MC–ICP–MS instruments because unlike the latter, they do not have a plasma source, which needs additional devices to power and cool the plasma, which in turn makes MC–ICP–MS more error-prone. (3) High resolution is not needed for TIMS because argon-related molecular interferences are irrelevant, making instrument tuning much easier. (4) The mass bias is more predictable and stable in MC–ICP–MS than in TIMS. However, mass bias can be corrected for each integration on both types of instruments by using the double spike technique, which typically lasts a few seconds. (5) MC–ICP–MS has greater sample throughput because it takes less time to measure one sample on the instrument (~15 minutes for MC–ICP–MS vs. 1–2 hours for TIMS). This advantage, however, may not be as significant as one might think for the following reasons: First, the isotopic results obtained by TIMS are relatively stable (no repeat is need), but repeated analyses are usually needed for MC–ICP–MS, as are more standard measurements. Second, instrument tuning time is much longer for MC–ICP–MS. Third, sample preparation can be a limiting factor. That said, with the development of automatic sample preparation systems (e.g., the prepFAST by Elemental Scientific) and optimization of sample purification procedures (e.g., Larsen et al. 2016), the high throughput offered by MC– ICP–MS could be highly advantageous for samples that do not require very high precision. The two inter-laboratory Cr isotope standards that have been used in most studies (both mass-independent and mass-dependent effects) are both from the National Institute of Standards and Technology (NIST): Standard Reference Material (SRM) 979, which has certified isotopic composition (50Cr/52Cr = 0.05186 ± 0.00010, 53Cr/52Cr = 0.11339 ± 0.00015,

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54

Cr/52Cr = 0.02822 ± 0.00006), and SRM 3112a, which does not have certified isotopic composition; however, the latter has been measured to have slightly lighter Cr isotopic composition than SRM 979 (Schoenberg et al. 2008; Shen et al. 2015). Both standards are supplied in Cr(III) form in nitric solution.

Notation Studies of mass-dependent and mass-independent Cr isotopic fractionation use different notation. “Mass-dependent” means that 53C/52Cr (y-axis) and 54C/52Cr (x-axis) are plotted on a straight line with a slope of ~0.5 (see Young et al. 2002 for details), whereas “massindependent” means that 53C/52Cr and 54C/52Cr deviate from this line. To express mass-dependent Cr isotopic fractionation relative to a laboratory standard, δ notation is used:   53Cr     52     Cr  smp  3 53 = δ Cr  53 − 1 10  Cr    52     Cr  std 

(1)

where the subscripts smp and std denote the Cr isotopic ratio of the sample and the standard, respectively. In some cases, it is appropriate to express the magnitude of isotopic fractionation between two phases a and b using a fractionation factor α: α=

Rb Ra

(2)

In the case of isotopic equilibrium, the extent of the equilibrium fractionation between a and b can be conveniently expressed as follows: Δ b − a = δ 53Crb − δ 53Cra ≈ 10 3 ln α

(3)

In the case of Cr reduction where isotope exchange between Cr(III) and Cr(VI) is limited because of the low solubility of Cr(III), Rb and Ra can be substituted by Rprod and Rreac, respectively, which are the 53Cr/52Cr ratios of the product Cr(III) at an instant in time and of the reactant Cr(VI) pool, respectively. The magnitude of kinetic fractionation can also be more conveniently expressed using ε notation:

ε=

(1 − α )10 3

(4)

For mass-independent isotopic measurements, the Cr isotopic composition is typically expressed in ε notation:   x Cr     52   Cr  smp  4  ε x Cr =  x − 1 10  Cr    52     Cr  std 

(5)

where x = 53, 54, and the subscripts smp and std indicate the Cr isotopic ratio in the sample and the standard, respectively. The isotopic ratios are corrected for the mass-dependent fractionation that occurs during measurements. To make this correction, the ratio 50Cr/52Cr

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is assumed to be natural. The difference between the measured value of 50Cr/52Cr and the natural ratio of the two isotopes is then taken as a measure of the instrumental mass fractionation, which is then used to correct 53Cr/52Cr and 54Cr/52Cr assuming the instrumental fractionation is mass-dependent and obeys exponential fractionation law (Maréchal 1999). We note that the notation ε has been used in both mass-dependent and massindependent Cr isotopic studies but has completely different meanings in the two contexts.

CHROMIUM ISOTOPE COSMOCHEMISTRY Chromium isotope cosmochemistry studies have mostly focused on the mass-independent effects of Cr isotopes, namely radiogenic effects on 53Cr (from the decay of short-lived 53Mn) and nucleosynthetic effects on 54Cr (also known as isotope anomalies). These two subjects have recently been reviewed, and readers are referred to recent reviews on short-lived chronometers by McKeegan and Davis (2014) and on nucleosynthetic isotope anomalies by Qin and Carlson (2016) for details. Extensive discussions on mass-independent effects are beyond the scope of this paper, but some of the important aspects are briefly reviewed here. 53

Mn–53Cr short-lived chronometer

Short-lived radionuclide chronometers. Short-lived nuclides were produced in supernovae before the formation of the solar system. They were accreted into the solar nebula and then the solar system; thus they were present at the very early stages of the Solar System. Briefly, for a short time (a few to tens of million years) in the early history of the solar system, 53Mn was still “alive”. During this time, differentiation of the solar materials led to varying Mn/Cr ratios, which ultimately led to different daughter 53Cr ingrowth. Excesses or deficits (deviation from the mass-dependent fractionation line) of 53Cr observed today provide evidence that differentiation events occurred prior to 53Mn becoming extinct. Various short-lived radionuclides are very useful tools for dating early solar system events that occurred within the first few million years to tens of millions of years of the formation of the solar system. A “fossil” isochron can be constructed for objects that were formed at the same time, and the slope corresponds to the initial parent isotope abundance at the time when the event occurred. By comparing this value with the initial abundance at the beginning of the solar system, the time elapsed after the formation of the solar system can be calculated. A “fossil” isochron can be constructed in a way similar to a traditional isochron except that on the x-axis, the parent isotope abundance is substituted with that of a stable isotope of the parent element because the parent isotope itself is extinct at the present day. It is important to note that unlike long-lived radioactive chronometers, short-lived radio chronometers can only date relative ages. To know absolute ages, one needs to have an age anchor for which both the absolute age (determined by long-lived radiochronometers) and the abundance of the short-lived nuclide at this age are known. Common age anchors are CAIs (calcium–aluminum-rich inclusions in chondrites, widely regarded as the oldest solids in the solar system) and angrites (differentiated achondrites, products of fast cooling; thus, the difference in closure time between various radiogenic chronometries is minimal). 53

Mn­–53Cr is one of the few short-lived chronometers that have achieved prominence (Birck and Allègre 1985, 1988; Rotaru et al. 1992; Lugmair and Shukolyukov 1998). Other examples are 26Al–26Mg, 146Sm–142Nd, and 182Hf–182W. The prominence of 53Mn–53Cr is partially due to the relatively high abundance of both Mn and Cr in most solar system objects. This system is very useful for dating processes that are related to volatility because of the different condensation temperatures of Mn and Cr [the 50% condensation temperatures for Cr and Mn are 1296 and 1158 K, respectively (Lodders 2003)]. The relatively long half-life of 53Mn of 3.7 Myr makes this chronometer suitable for dating events from nebular processes to the formation and differentiation of early planetesimals.

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Evidence of extant 53Mn has been reported for various solar system objects including CAIs, chondrules, and bulk and components of chondrites and achondrites (Birck and Allègre 1985, 1988; Lugmair and Shukolyukov 1998; Nyquist et al. 2001; Trinquier et al. 2008b). One of the complications of using this chronometer in early days, however, was that there were debates about the homogeneity in the initial distribution of 53Mn in the solar system (e.g. Lugmair and Shukolyukov 1998; Nyquist et al. 2001). However a recent study showed that 53Mn was initially homogeneously distributed in the inner Solar System (Trinquier et al. 2008b). CAIs are the obvious target to obtain initial 53Mn, but they contain very small amounts of Mn and Cr because both are moderately and differentially volatile (Lodders 2003). The most appropriate age anchors for Mn–Cr system are angrites. The initial 53Mn/55Mn ratio was determined for angrite D’Orbigny ((3.24 ± 0.04)×10−6; Glavin et al. 2004) and LEW 86010 ((1.25 ± 0.07) × 10−6; Lugmair and Shukolyukov 1998). Combining the absolute Pb–Pb ages of these two angrites (4563.37 ± 0.025 Ma for D’Orbigny and for 4558.62 ± 0.15 Ma for LEW 86010; Brennecka and Wadhwa 2012), the Mn–Cr and U–Pb ages calculated for various solar system objects are mostly concordant. 54

Cr anomalies

Heavy elements (> He) in the solar system were synthesized before the solar system formed. Different nucleosynthetic sources produce elements with distinct isotopic signatures. Isotope anomalies are mass-independent isotopic variations that can be explained by different relative contributions of the nucleosynthetic processes that made all the heavy elements in the solar system. Isotope anomalies recorded in meteorites are very useful for tracing the sources and distributions of nucleosynthetic products in the early solar nebula and for understanding the stellar processes that made the elements. Recently, the discovery of planetary-scale isotope anomalies has provided a viable means for tracing the astrophysical environments of solar system formation, the mixing processes of nuclides from different sources, and the generic relationship between different meteorite groups. 54 Cr is a neutron-rich isotope, along with 48Ca, 50Ti, 62Ni, and 64Ni from other iron peak elements, which are thought to be produced in similar astrophysical environments, but the site and method of their production are still somewhat uncertain (Meyer et al.1996; Woosley and Heger 2007; Wasserburg et al. 2015). Anomalies in 54Cr (i.e. non-zero ε54Cr) were first found in Allende CAIs, with calcium-aluminum-rich inclusions with fractionation and unknown nuclear effects (FUN CAIs) exhibiting diverse ε54Cr values from –151 to 48 parts per ten thousand, compared with ~7 for normal CAIs (Birck and Allègre 1984; Papanastassiou 1986). 54 Cr anomalies were also revealed during stepwise acid digestion of chondrites: The fractions dissolved by early, weak acid leaches are characterized by negative 54Cr anomalies. Later, stronger acid leachates and the acid residues have positive anomalies (Rotaru et al. 1992; Podosek et al. 1997; Trinquier et al. 2008b; Qin et al. 2011a); this suggests heterogeneous mixing of products from different nucleosynthetic sources within carbonaceous chondrites.

At the whole-meteorite scale, different groups of meteorites show a range in 54Cr/52Cr of approximately 2 parts per ten thousand. The variability is very systematic and therefore can be used for distinguishing meteorite classes (Table 2). For instance, carbonaceous chondrites display ε54Cr values ranging from 0.4 to 1.6, compared with enstatite chondrites, which have similar ε54Cr values (0) to those of terrestrial rocks; ordinary chondrites exhibit a constant deficit of ~0.4; achondrites, iron meteorites, and Martian meteorites show deficits of ~0.2 to 0.9 in ε54Cr; lunar rocks have the same ε54Cr values as terrestrial rocks (Shukolyukov and Lugmair 2006; Trinquier et al. 2007; Qin et al. 2010a,b; Yamakawa et al. 2010; Warren 2011). The carrier phases of 54Cr were identified as sub-micron oxide grains from a Type II supernova, and the distribution of 54Cr anomalies among different meteorite groups is thought to result from temporally or spatially heterogeneous distribution of these carrier phases (Dauphas et al. 2010; Qin et al. 2011b).

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Table 2. 54Cr Isotopic anomalies in bulk meteorites. Material type Terrestrial Carbonaceous chondrites

ε54Cr (parts per ten thousand) 0 0.4–1.6

Enstatite chondrites

0

Ordinary chondrites

−0.4

Achondrites Iron meteorites

~ −0.2 to −0.9

Martian meteorites Lunar rocks

0

Data sources: Shukolyukov and Lugmair (2006); Trinquier et al. (2007); Qin et al. (2010a,b); Yamakawa et al. (2010); Warren (2011)

Anomalies of other neutron-rich isotopes have also been found, and these anomalies are more or less correlated with each other at the bulk-meteorite scale (Regelous et al. 2008; Trinquier et al. 2009; Warren 2011; Dauphas et al. 2014), suggesting similar causes for these isotope anomalies. It is important to note that most terrestrial rocks do not exhibit variations in ε53Cr and ε Cr. However, if they do, it is taken as evidence of meteoritic impact (Trinquier et al. 2006). 54

CHROMIUM ISOTOPIC FRACTIONATION IN HIGH-TEMPERATURE SETTINGS Bulk silicate earth and meteorites Compared with low-temperature processes (discussed below), the stable Cr isotopic fractionation induced by high-temperature processes is poorly constrained. Schoenberg et al. (2008) first studied the Cr isotopic compositions of a set of mantle-derived rocks from different sources, including mantle xenoliths, ultramafic rocks and cumulates, and oceanic and continental basalts. The investigated samples gave a limited δ53Cr range of −0.211‰ to −0.017‰, with an average of −0.124 ± 0.101‰, implying a relatively homogeneous mantle source in regard to Cr isotopic compositions and limited Cr isotopic fractionation during partial melting and magma differentiation. Farkaš et al. (2013) observed that igneous chromites have slightly heavier Cr isotopic compositions (with an average δ53Cr value of −0.079 ± 0.129‰) than the average value of the bulk silicate earth (BSE) reported by Schoenberg et al. (2008) and then inferred that chromite-bearing mantle might be isotopically heavier than mantle without chromite. This finding was confirmed by Shen et al. (2015), who directly compared the chromite-bearing mantle and chromite-free mantle. Lunar mare basalts show Cr isotopic variation correlated with MgO, likely due to crystallization of spinel (Bonnand et al. 2016). Moynier et al. (2011) found that chondrites from all subgroups have systematically lower δ53Cr, ranging from −0.4 to −0.2‰ (Moynier et al. 2011), than the silicate earth (Schoenberg et al. 2008). This was interpreted by Moynier et al. (2011) as a result of incorporation of light Cr into the core. This finding was based on first-principle calculation results, which showed that potential Cr-bearing phases (metal or sulfide in the core) have light Cr isotopic composition compared with co-existing Cr-bearing minerals in the silicate, with Cr in the former dominated by Cr0 and Cr2+ and in the latter dominated by Cr3+ (Moynier et al. 2011), given that the condition of the proto-Earth is sufficiently oxidizing to contain a large portion of Cr3+. However, the light Cr isotopic data for chondrites have not been reproduced by several recent studies, which showed instead that chondrites have similar δ53Cr values to those of Earth rocks (Qin et al. 2015; Bonnand et al. 2016; Schoenberg et al. 2016).

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We note that the first-principle calculation results for Cr isotopes assume that Cr is a major element, likely because of limitations on the computing power of modern computers (Moynier et al. 2011). However, Cr is a trace element in most major mantle minerals. Further theoretical calculations and measurements on major minerals from the mantle will provide important insights into Cr isotopic behavior during partial melting and fractional crystallization. Shen et al. (in preparation) have performed both ionic modeling and measurements on the Cr isotopic composition of major mantle minerals. Both results indicate that the δ53Cr value decreases slightly in the order of spinel, chromite > clinopyroxene, orthopyroxene > olivine.

Serpentinization and metamorphism Recent experiments and natural observations revealed the high mobility of Cr(III) in Cl-rich fluids (Ottaway et al. 1994; Klein-BenDavid et al. 2009, 2011; Spandler et al. 2011; Watenphul et al. 2014), which are common during serpentinization and oceanic crust dehydrations. High Cr mobility allows the opportunity for Cr isotopic fractionations, which in turn permits the study of subduction and crust–mantle recycling. Recently, several studies have been focused on Cr isotopic fractionations during serpentinization and metamorphic dehydration. Farkaš et al. (2013) first reported extremely high δ53Cr in serpentinites (up to ~+1.22‰ δ53Cr), and positive correlations between δ53Cr values and various alteration indexes. Thus, they proposed that serpentinization could shift altered peridotites to isotopically high δ53Cr, which was interpreted as a result of reduction of isotopically heavier Cr(VI) in serpentinizing fluids. Subsequently, Wang et al. (2016c) presented detailed Cr isotopic investigations of a series of serpentinites and found that the altered peridotites with lower Cr contents had higher δ53Cr values. Two possibilities were proposed: (i) Light Cr isotopes were lost to fluids during serpentinization as a result of kinetic fractionation, leaving isotopically heavy Cr behind; or (ii) Cr was initially lost to fluids without isotopic fractionation, then isotopically heavy Cr was accumulated during later-stage sulfate reduction. Although the specific Cr isotopic fractionation mechanism remains controversial, the consistent observations of isotopically heavy serpentinites imply potentially significant Cr isotopic fractionation during dehydration accompanying subduction. To balance the isotopically heavy serpentinites, isotopically light-Cr phases are yet to be identified. Some isotopically heavy metamorphic minerals have been reported by Farkaš et al. (2013), but no systematic Cr isotopic fractionation was observed in subduction-related metamorphosed mafic rocks from the Dabie-Sulu orogen (Shen et al. 2015; Wang et al. 2016c). These metamorphic rocks were thought to have been subducted to ultra-high P–T conditions and then exhumed to low P–T conditions (Zheng 2008; Wang et al. 2013). They span greenschists, amphibolites, and eclogites, yet their δ53Cr values are indistinguishable from the BSE range reported by Schoenberg et al. (2008). The contrasting δ53Cr between serpentinized peridotites and metamorphosed mafic rocks is enigmatic but interesting because it motivates the continued search for the missing isotopically light counterparts in the subduction system. One explanation for the absence of Cr isotopic variation in some metamorphic processes could be a lack of fluid, which hinders Cr mobility and thus limits Cr isotopic fractionation (Shen et al. 2015). Geochemical indices for fluid activities can be utilized to test this hypothesis. Finally, the Cr isotopic compositions of different terrestrial reservoirs are summarized in Figure 2. The reader will notice that weathered rocks are the only geological reservoir possessing negative δ53Cr. Therefore, our current terrestrial Cr isotope inventory may be skewed to the positive side, and additional future measurements are needed for a more complete picture regarding Cr isotope inventory on Earth.

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δ53Cr -4

-2

0

2

4

6

8

igneous rocks sedimentary rocks soil, weathering profile river water seawater groundwater serpentinized oceanic crust Figure 2. Cr isotopic composition of different terrestrial reservoirs.

MECHANISMS OF Cr ISOTOPIC FRACTIONATION IN LOW-TEMPERATURE SETTINGS Owing to the contrasting geochemical behavior between insoluble Cr(III) and soluble Cr(VI), reduction of Cr(VI) provides a viable means to remediate Cr(VI) contamination of water sources (Blowes et al. 2000; Crane and Scott 2012). Because substantial isotopic fractionation occurs during reduction, the Cr isotopic composition of groundwater samples is very useful for monitoring and potentially quantifying the extent of reduction. Furthermore, the reduction-associated isotopic fractionation in ancient sediments has been used to track the redox evolution of Earth’s surface environment over geological timescales (e.g., Frei et al. 2009; Crowe et al. 2013, Planavsky et al. 2014). Given the great interest in using Cr isotopes in environmental and paleo-redox studies, various reduction mechanisms (described below) have been subjected to extensive studies because they can potentially generate distinctive Cr isotopic fractionations. The results of these experiments are summarized in Table 3; these results can be used for fingerprinting these processes.

Reduction Abiotic reduction. Sulfides, ferrous iron, organic matter and other organic reductants in sediments and in surface and groundwater can serve as natural Cr(VI) reductants (Fendorf and Li 1996; Patterson et al. 1997; Pettine et al. 1998; Kim et al. 2001; Wielinga et al. 2001; Graham and Bouwer 2009; Kitchen et al. 2012). Hexavalent Cr can also be reduced by granular zerovalent iron (ZVI) particles during remediation (Alowitz and Scherer 2002; Lee et al. 2003). The reduction by ZVI often takes place in permeable reactive barriers, where ZVI is used to reduce

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Table 3. Chromium isotopic fractionation (δproduct – δreactant) induced by different mechanisms determined by laboratory experiments. [continued on next page] Reagent

Isotopic fractionation

References

magnetite, sediment slurries

−3.5 ± 0.1‰

Ellis et al. (2002)

Sediment slurry

−2.4 ± 0.1‰ to −3.1 ± 0.1‰

Berna et al. (2010)

Fe(II)-doped goethite

−3.91 ± 0.16‰

Basu and Johnson (2012)

FeS

−2.11 ± 0.04‰

Green rust

−2.65 ± 0.11‰

Siderite

−2.76 ± 0.25‰

Hanford sediments

−3.27 ± 0.12‰

Geobacter sulfurreducens

−3.03 ± 0.12‰

Shewanella sp. (NR)

−2.17 ± 0.22‰

Pseudomonas stutzeri DCP-Ps1

−3.14 ± 0.13‰

Desulfovibrio vulgaris

−3.14 ± 0.11‰

Desulfovibrio vulgaris (aerated)

−2.78 ± 0.27‰

Pseudomonas stutzeri RCH2 (aerated)

~−2‰

Pseudomonas stutzeri RCH2 (denitrifying)

~−0.4‰

Bacillus sp. QH−1 with glucose

−2.00 ± 0.21‰

Bacillus sp. QH−1 without glucose

−3.74 ± 0.16‰

Bacillus sp. QH−1, T = 4 °C

−7.62 ± 0.36‰

Bacillus sp. QH−1 glucose, T = 15 °C

−4.59 ± 0.28‰

Bacillus sp. QH−1 glucose, T = 25 °C

−3.09 ± 0.16‰

Reduction

Bacillus sp. QH−1 glucose, T = 37 °C

Basu et al. (2014)

Han et al. (2012) Xu et al. (2015)

−1.99 ± 0.23‰ −

Shewanella oneidensis strain MR−1 low e donor

−4.21 ± 0.38‰

Shewanella oneidensis strain MR−1 high e− donor

−1.8 ± 0.2‰

Dissolved Fe(II) in batch mode

−3.6‰

Sikora et al. (2008) Døssing et al. (2011)

Dissolved Fe(II) in constant addition mode with green −1.5‰ rust being formed Dissolved Fe(II), pH 4 to 5.3

−4.20 ± 0.11‰

Kitchen et al. (2012)

Organic acids (Elliot fulvic acid, Waskish humic acid, −3.11 ± 0.11‰ Mandelic acid Notes: The “-”and “+” signs of fractionation denote that the product is isotopically lighter and heavier than the reactant, respectively.

Cr(VI) (Blowes et al. 2000; Crane and Scott 2012). ZVI particles can reduce Cr(VI) directly, or they can react with H + to form ferrous iron, which then reduces Cr(VI). The reduction of Cr(VI) by organic matter also requires the incorporation of H+ (Jamieson-Hanes et al. 2012b).

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Table 3 (cont’d). Chromium isotopic fractionation (δproduct – δreactant) induced by different mechanisms determined by laboratory experiments. Reagent

Isotopic fractionation

References

Adsorption γ-Al2O3, goethite

< −0.04‰*

Ellis et al. (2004) Oxidation

Pyrolusite (β-MnO2)

≤ +1‰

Ellis et al. (2008)

Birnessite (δ-MnO2)

−2.5‰ to + 0.7‰

Bain and Bullen (2005)

Birnessite (δ-MnO2)

−0.5‰ to 0.0‰

Wang et al. (2010)

H2O2

~ +0.2‰ to 0.6‰

Zink et al. (2010)

Aqueous Cr(III)–Cr(VI) isotope exchange No reductant/oxidant

5.8 ± 0.5‰, Cr(VI) isotopically heavier than Cr(III)**

Wang et al. (2015)

Cr(VI) uptake by abiotic calcite Fast precipitation

+0.06‰ to + 0.18‰

Slow precipitation

+0.29 ± 0.08‰

Rodler et al. (2015)

Cr uptake by coral aragonite ~ −1.4‰ to −0.6‰

Pereira et al. (2015)

Notes: The “-”and “+” signs of fractionation denote that the product is isotopically lighter and heavier than the reactant, respectively. *This is true only at adsorption–desorption equilibrium. Under disequilibrium conditions, larger but transient isotopic fractionation could occur. **Experimentally determined equilibrium fractionation is in line with the theoretical estimate (Schauble et al. 2004).

Biotic reduction. Biotic Cr(VI) reduction is one of the most important processes in the Cr biogeochemical cycle. Hexavalent Cr can be naturally reduced by microbes in aquifers. Because Cr(VI) is a toxic species and often does not directly participate in microbial metabolism, it is typically reduced through co-metabolism; that is to say, bacteria do not use Cr for energy, and Cr is reduced by a variety of reducing compounds found in living cells. However, in reducing environments such as some organic-rich sediment packages within aquifers where oxygen is depleted, bacteria such as Shewanella oneidensis can use Cr(VI) instead of oxygen for respiration (Middleton et al. 2003). Specifically, S. oneidensis oxidizes organic matter to extract energy, a process that produces electrons. The bacteria then transfer the electrons to extracellular Cr(VI) using c-type cytochromes (Myers and Myers 2000). In other scenarios, CrO42− could mistakenly enter cells because of the structural similarity between CrO42− and SO42−. Once within the cell membrane, Cr(VI) is reduced via the sulfate reduction mechanism (Viti et al. 2009). Chromium can also be reduced by chromate reductases, which use NADH or NADPH as cofactors under aerobic conditions (Park et al. 2000). Because of the Cr(VI)-reducing capabilities of microbes, bio-stimulation—injection of electron donors to stimulate microbial growth—is considered an efficient way to enhance Cr(VI) immobilization. Because many microbes are native to the environment, bio-remediation has proven to be efficient and inexpensive, and it usually causes no secondary pollution to the environment (e.g., Palmer and Wittbrodt 1991; Zayed and Terry 2003). In addition to anaerobic microbes in aquifer systems, aerobic microbes have also been utilized to immobilize Cr(VI) in industrial wastewater (Ohtake et al. 1990; Ackerley et al. 2004; Xu et al. 2015). Reduction-induced Cr isotopic fractionation. Because Cr(VI) usually exists in the form of CrO42− or HCrO4−, the reduction of Cr(VI) involves the breaking of a Cr–O bond. Because

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light Cr isotopes have higher vibrational frequencies, their bonds with O are easier to break than those of heavy isotopes, resulting in greater reaction rates for lighter Cr isotopes and thus in their enrichment in the reduction product Cr(III) (Schauble 2004). The solid Cr(III) is then physically separated from soluble Cr(VI) to prevent backward reaction. Therefore, during reduction, the Cr(III) produced at each moment in time is constantly offset from the isotopic composition of the remaining Cr(VI) pool by a kinetic isotope fractionation factor ε (a negative value), which varies widely depending on the reaction mechanisms and environmental conditions (Table 3). In a closed aqueous system, isotope fractionation during reduction of Cr(VI) follows a Rayleigh fractionation model

(

)

 δ53Crini + 103 f ( a −1)  − 103 δ53Cr =  

(6)

where δ53Cr and δ53Crini refer to the unreacted Cr(VI) pool at the time of sampling and at the start of the reaction, respectively; f refers to the fraction of Cr(VI) remaining in the solution. To use the Rayleigh fractionation model, three conditions must be met: (i) the system is well mixed and closed; (ii) all of the reduction product Cr(III) is sufficiently removed from the solution, and/or no further isotopic exchange between Cr(VI) and Cr(III) occurs; and (iii) the kinetic isotopic fractionation factor α does not change with time. Although Cr(III) is usually insoluble in the natural pH range, it can be solubilized by complexation with organic matter. Fortunately, as will be shown later, the exchange timescale between Cr(VI) and Cr(III) is relatively long compared with most experimental as well as remediation timescales. Therefore, under controlled laboratory conditions, these three conditions are usually met, and the fractionation factor can be derived from the correlation diagram between Cr isotopic composition and the concentration of Cr(VI) remaining in the solution at different time points (Fig. 3). Batch and column (flow-through) Cr reduction experiments performed under a wide range of conditions with various abiotic reductants including magnetite, FeS, dissolved Fe(II), and natural sediments obtained ε values ranging from 0.2 to 5, with most values falling within the range of 2‰ to 4‰ (Ellis et al. 2002; Berna et al. 2010; Zink et al. 2010; Døssing et al. 2011;

Figure 3. δ53Cr in the remaining Cr(VI) pool vs. the fractionation of Cr(VI) remaining in a Rayleigh fractionation model during Cr(VI) reduction (0‰ at start) with an ε value of 3. The ε value is obtained by fitting the Rayleigh equation (Eqn. 7) to the data. Conversely, if ε is known, one can estimate the extent of reduction (i.e., the fraction of Cr(VI) remaining) by measuring the δ53Cr of Cr(VI) in solution.

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Basu and Johnson 2012; Jamieson-Hanes et al. 2012b, 2014; Kitchen et al. 2012) (Fig. 4). Small ε values are usually associated with fast Cr(VI) reduction. The smallest ε values were observed in flow-through experiments with ZVI as the reductant (ε values ranged from 0.2 to 1.5; Jamieson-Hanes et al. 2012b). Similarly, kinetic isotopic fractionation induced by biotic Cr(VI) reduction varies widely (Table 3). For instance, under anaerobic conditions, higher donor concentrations were linked to faster Cr reduction accompanied by lower ε values, compared with that at low donor concentrations (Sikora et al. 2008), but donor concentration does not seem to have an effect on isotopic fractionation under aerobic conditions (Xu et al. 2015). In addition, ambient redox conditions also seem to have an effect. For instance, Han et al. (2012) found that under denitrifying conditions, there is much less fractionation by enzymatic reduction of Cr(VI) than under aerobic conditions, even though the reduction rates are similar. Furthermore, Xu et al. (2015) also found an inverse relationship between temperature and ε during microbial Cr(VI) reduction. Reduction of Cr under comparable experimental conditions by different types of microbes did not generate a large effect on the isotopic fractionation factor, except in one species (Basu et al. 2014). Chromium isotopic fractionation during plant uptake was investigated by Ren et al. (2015); the authors used an enriched tracer and showed that light Cr isotopes were enriched in the leaves, whereas the roots were enriched in heavy isotopes. As for the field studies, the ε values derived from field studies are smaller than those obtained in laboratory experiments with the same reductant by a factor of at least 2 (e.g. Berna et al. 2010).

 

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Reviews in Mineralogy & Geochemistry Vol. 82 pp. 511-542, 2017 Copyright © Mineralogical Society of America

The Isotope Geochemistry of Ni Tim Elliott Bristol Isotope Group School of Earth Sciences University of Bristol Bristol BS8 1RJ UK [email protected]

Robert C. J. Steele Institute for Geochemistry and Petrology ETH Zürich Zürich 8092 Switzerland [email protected]

INTRODUCTION Nickel is an iron-peak element with 5 stable isotopes (see Table 1) which is both cosmochemically abundant and rich in the information carried in its isotopic signature. Significantly, 60Ni is the radiogenic daughter of 60Fe, a short-lived nuclide (t1/2 = 2.62 Ma; Rugel et al. 2009) of a major element. 60Fe has the potential to be both an important heat source and chronometer in the early solar system. 60Ni abundances serve to document the prior importance 60 Fe and this is a topic of on-going debate (see Extinct 60Fe and radiogenic 60Ni). The four other stable Ni nuclides span a sizeable relative mass range of ~10%, including the notably neutronrich nuclide 64Ni. The relative abundances of these isotopes vary with diverse stellar formation environments and provide a valuable record of the nucleosynthetic heritage of Ni in the solar system (see Nucleosynthetic Ni isotopic variations). Ni occurs widely as both elemental and divalent cationic species, substituting for Fe and Mg in common silicate structures and forming Fe/Ni metal alloys. The Ni isotope chemistry of all the major planetary reservoirs and fractionations between them can thus be characterized (see Mass-Dependent Ni isotopic Variability). Ni is also a bio-essential element and its fractionation during low-temperature biogeochemical cycling is a topic that has attracted recent attention (see Mass-Dependent Ni isotopic Variability).

Notation Much of the work into Ni has been cosmochemical, focussing on the nucleosynthetic origins of different meteoritic components. Such studies have primarily investigated mass-independent isotopic variations, both radiogenic and non-radiogenic, which require choosing a reference isotope pair for normalization. Throughout this work we use 58Ni–61Ni as the normalizing pair, in keeping with current practice in the field. An alternative 58Ni–62Ni normalization scheme has previously been used for bulk analyses (Shimamura and Lugmair 1983; Shukolyukov and Lugmair 1993a,b; Cook et al. 2006, 2008; Quitté et al. 2006, 2011; Chen et al. 2009) and one early study used 58Ni–60Ni (Morand and Allègre 1983). Although the large isotopic variability accessible by in situ analyses often makes external normalization a viable option for mass-independent measurements by secondary ionization mass-spectrometry (SIMS), some have employed internal 1529-6466/17/0082-0012$05.00

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Table 1. Ni isotopic abundances (Gramlich et al. 1989), nuclide masses (Wang et al. 2012) and atomic weight (Wieser et al. 2013). Ni isotopic abundances also expressed as atomic ratios. 58

60

Ni

61

Ni

62

Ni

64

Ni

Ni

Atomic Fraction

0.68076886

0.26223146

0.01139894

0.03634528

0.00925546

2 SE

5.92 × 10−5

5.14 × 10−5

4.33 × 10−5

1.14 × 10−5

5.99 × 10−5

Nuclide Mass (amu)

57.93534241 59.93078589 60.93105557 61.92834537 63.92796682

IUPAC Atomic weight (amu)

58.6934 ± 0.0004 60

58

Ni/ Ni

Atomic Ratio

61

Ni/58Ni

62

Ni/58Ni

64

Ni/58Ni

0.385198965 0.016744215 0.053388576 0.013595598 8.27 × 10−5

2 SE

6.52 × 10−6

1.74 × 10−5

8.88 × 10−6

normalization in determinations of 60Ni/61Ni (Tachibana and Huss 2003; Tachibana et al. 2006; Mishra and Chaussidon 2014; Mishra et al. 2016). Given unresolvable Fe and Zn interferences on masses 58 and 64, this requires normalizing to 62Ni/61Ni. In this review, all data have been renormalized to 58Ni–61Ni, where possible. Some studies have only reported normalized data and so such conversion cannot be made. Fortunately, the subtle differences resulting from different normalizations do not affect the inferences being made in these cases and we simply indicate the normalization scheme used. To be clear about these potentially important details, we use a notation proposed by Steele et al. (2011), which includes this information. For example:

(

= ε60/58 Ni 58/61

60

)

Ni / 58 Nisample / 60 Ni / 58 Nistandard − 1 × 10000 norm 58/61 norm 58/61

(1)

or the parts per ten thousand variation of 60Ni/58Ni (internally normalized to a reference 58 Ni/61Ni) relative to a standard measured in the same way. The established isotopic standard for Ni is the National Institute of Standards and Technology Standard Reference Material (NIST SRM) 986, Gramlich et al. (1989). Reference Ni isotope ratios for this standard are reported in Table 1. This NIST SRM has been widely used, providing a valuable common datum in all but the earliest work. If it is necessary to clarify which reference standard has been used, the notation above can be augmented, e.g. ε60Ni58/61 (NIST 986). For elements such as Ni, however, where the same standard is conventionally used, we feel this additional information can be omitted without too much confusion, provided it is imparted elsewhere (as we do here). We use the epsilon notation solely for mass-independent isotopic data (internally normalized). This approach is typical although not universal and the presence of the subscript in our notation (Eqn. 1) makes the use of internal normalization evident. We report mass-dependent variations in the delta notation: = δ60 Ni

(

60

)

Ni/ 58 Nisample / 60 Ni/ 58 Nistandard − 1 × 1000

(2)

As for the mass-independent work, NIST SRM 986 is extensively used as the Ni isotope reference standard in all studies other than Moynier et al. (2007), and is implicit in Equation (2). In Equation (2), we follow another proposal made in Steele et al. (2011) to report the isotope ratio used, i.e. δ60/58Ni instead of δ60Ni. This removes any ambiguity over which nuclide is used as the denominator. For an element such as Ni, with more than two stable isotopes, such qualification is valuable. We suggest this systematic notion could be useful more generally. Moynier et al. (2007) reported their mass-dependent Ni isotope data as dNi, an error weighted, average fractionation per unit mass difference, using the three measured ratios (60Ni/58Ni)/2, (61Ni/58Ni)/3 and (62Ni/58Ni)/4. This is an interesting idea (see also Albalat et

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al. 2012), which reduces the error for the sample-standard bracketing technique by using all measured data. The alternative method for determining mass-dependent isotopic fractionation is by double-spiking (see review by Rudge et al. 2009). Double-spiking requires measurements of four isotopes, thus yielding three independent isotope ratio determinations. Given a troublesome Zn interference on mass 64, all double-spiked studies to date have used a 61Ni–62Ni double spike and employed 58Ni, 60Ni, 61Ni and 62Ni in the data reduction. With ostensible similarity to the sample-standard bracketing approach of Moynier et al. (2007), the combined measurements of these four isotopes yield a single value of natural isotopic fractionation, which is normally converted into a more tangible delta value for a specific but arbitrary isotope ratio (e.g. δ60/58Ni). The key difference in double-spiking is that the additional isotope ratios are used to constrain explicitly instrumental mass-bias. Namely this procedure improves accuracy, whereas in sample-standard bracketing instrumental mass bias is assumed to be identical for sample and standard and the additional isotope measurements are used to improve precision. Both methods described above use measurements of 60Ni for determining mass-dependent Ni isotope variability. It is germane to consider whether or not it makes good sense to use a radiogenic isotope for such a purpose. For terrestrial samples, there should be no variability in the relative abundance of radiogenic 60Ni, given likely terrestrial isotopic homogenization after parental 60Fe became extinct. For extra-terrestrial samples, this is potentially a consideration, but bulk variations in ε60Ni58/61 are typically small (~0.1), dominantly nucleosynthetic rather than radiogenic and associated with mass-independent variability of other isotopes (see the section Nucleosynthetic Ni isotopic variations). For the most accurate mass-dependent measurements, a second mass-independent isotopic determination is therefore required (e.g. Steele et al. 2012), but such mass-independent variability does not have a significant impact on more typical massdependent determinations at the delta unit level (see the section Magmatic Systems). In the section Extinct 60Fe and radiogenic 60Ni we address the presence of 60Fe in the early solar system from 60Ni measurements of meteoritic samples. Such determinations yield initial 60Fe/56Fe for the objects analyzed, denoted 60Fe/56Fe°. Given samples that yield precise values of 60Fe/56Fe° may have different ages, it is useful to calculate 60Fe/56Fe° at a common reference time, typically the start of the solar system as marked by calcium aluminium rich inclusion formation. Such a solar system initial value is abbreviated to 60Fe/56Fe°SSI. Throughout this review, the uncertainties quoted for various average measurements are two standard errors, unless otherwise stated.

NUCLEOSYNTHETIC Ni ISOTOPIC VARIATIONS Nickel is significant element in stellar nucleosynthesis. Nickel-62 has the highest binding energy per nucleon of any nuclide; no nuclear reaction involving heavier nuclides can produce more energy than it consumes. It is often, incorrectly, said that 56Fe has the highest binding energy per nucleon, likely due to the anomalously high abundance of 56Fe. In fact, 56 Fe is dominantly produced in stars as the decay product 56Ni, which is the result of the last energetically favorable reaction during Si burning. It is thought the Ni isotopes are dominantly produced during nuclear statistical equilibrium (NSE or the e-process) in supernovae (Burbidge et al. 1957). There are two main astrophysical environments in which the majority of Ni is thought to be produced, these are the type Ia (SN Ia) and type II (SN II) supernovae. SN Ia are thought to be the violent explosions of carbon– oxygen white dwarves which accrete material from a binary host to reach the Chandrasekhar limit (1 mm), nucleosynthetic mixtures that contrast with the bulk solar system. Following reports of massindependent variations of other iron-peak nuclides (e.g. Lee et al. 1978; Heydegger et al. 1979), the first Ni isotopic analyses of Allende CAIs (Morand and Allègre 1983; Shimamura and Lugmair 1983) failed to resolve signatures that differed from terrestrial values, in all but a single, highly anomalous ‘FUN’ inclusion (Shimamura and Lugmair 1983). Subsequent improvements in precision allowed Birck and Lugmair (1988) to resolve excesses of ~1 ε62Ni61/58 and ~3 ε64Ni61/58 within Allende CAIs (Fig. 1), which they noted was in keeping with a neutron-rich, equilibrium process nucleosynthesis. These findings were pleasingly compatible with anomalies in neutron-rich isotopes of Ca (Jungck et al. 1984) and Ti (Heydegger et al. 1979; Niederer et al. 1980; Niemeyer and Lugmair 1981) from previous studies and excesses of 54Cr in their own work (Birck and Lugmair 1988). This landmark contribution identified the key mass-independent variations in Ni isotopes that would subsequently become apparent in bulk meteorite analyses. The CAIs analysed by Birck and Lugmair (1988) also displayed 60Ni enrichments, ε60Ni61/58 ~ 1 (Fig. 1a), potentially related to the decay of 60Fe co-produced with the neutronrich Ni isotopes (see Extinct 60Fe and radiogenic 60Ni). If these ε60Ni61/58 values are taken solely as the consequence of in situ 60Fe decay, they imply initial 60Fe/56Fe ~ 1 × 10 − 6, but Birck and Lugmair (1988) cautioned against such an inference, given associated nucleosynthetic variations of comparable magnitude. Further analyses of CAIs by Quitté et al. (2007) similarly showed positive ε60Ni61/58 and ε62Ni61/58 (see Fig. 1a); the method used in this study suffered from too large 64Zn interferences to make precise ε64Ni61/58 measurements. The authors attributed their observations to synthesis of 60Fe and 62Ni (and 96Zr) in a neutron burst event, followed by the decay of 60Fe. Birck and Lugmair (1988) had argued against this style of model, given the absence of predicted, associated 46Ca anomalies in CAIs. However, Quitté et al. (2007) tentatively inferred 60 Fe/56Fe°SSI > 1 × 10 − 6 from their nucleosynthetic model, a two point internal CAI isochron and the interpretation of ε60Ni61/58 excesses in two CAIs without ε62Ni61/58 anomalies.

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3 2

515

Calcium, aluminium rich inclusions Carbonaceous Chondrites Ordinary Chondrites Enstatite Chondrites Iron meteorites Birck and Lugmair Cook et al. Quitte et al.

1 0 −1 −2 −4

(a) −3

−2

−1

0

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ε62Ni58/61

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8 6

ε64Ni58/61

4 2 0 −2 −4 −6 −4

(b) −3

−2

−1

0

1

ε62Ni58/61

2

3

4

Figure 1. Mass independent nickel isotope data for meteorites and CAIs from earlier studies, (a) ε60Ni58/61 vs. ε62Ni58/61 and (b) ε64Ni58/61 vs. ε62Ni58/61. Data from Birck and Lugmair 1988; Cook et al. 2006, 2008; Quitté et al. 2006, 2007. The dashed lines indicate the vector of change in composition (from the origin) as a result of error or interference (moving to the lower left) in 61Ni. Some scatter appears to be along this trajectory. For ready comparison with the higher precision data, the blue boxes indicate the dimensions of parameter space shown in Figure 4. The black line with grey band shows an extrapolation of a least squares York regression and the 2σ uncertainty error envelope (York 1969; Mahon 1996) through the bulk meteorite and peridotite data of Steele et al. (2012), see Figure 4. This trend is shown in Figure 1b for reference and notably passes through the CAI data.

Better precision was required to investigate mass-independent Ni isotopic variability between bulk meteorite samples. This came with improvements in mass-spectrometry, including multi-collection systems and their coupling with plasma sources (MC–ICPMS). The latter allows intense beams to be runs with relative ease, permitting counting statistical limitations to be overcome given sufficient sample availability. Nonetheless, the technique does require careful monitoring of a wide range of potential sample and plasma related interferences that may be significant at high precision (see common interferences listed in Quitté and Oberlei 2006 and Steele et al. 2011). Some of the earlier MC–ICPMS studies focussed on iron meteorites or the metallic phases of iron-bearing chondrites (Cook et al. 2006; Quitté et al. 2006; Moynier et al.

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2007; Dauphas et al. 2008). This approach usefully exploits the natural concentration of Ni in a form readily purified to provide sufficient material for high precision analysis. In comparison, work on silicate samples requires more involved separation of Ni from a wider range of elements, typically accomplished using the highly Ni-specific complexing agent dimethylglyoxime either in solvent extraction (e.g. Morand and Allègre 1983; Shimamura and Lugmair 1983), mobile phase (e.g. Wahlgreen et al. 1970; Victor 1986; Steele et al. 2011; Gall et al. 2012, Chernonozhkin et al. 2015) or stationary phase of ion-chromatography (e.g. Quitté and Oberli 2006; Cameron et al. 2009). Alternatively, cation chromatography using a mixed HCl–acetone eluent (Strelow et al. 1971) has also been successfully used (e.g. Tang and Dauphas 2012). Initial MC–ICPMS studies on meteoritic metal phases (Cook et al. 2006; Quitté et al. 2006; Moynier et al. 2007; Dauphas et al. 2008) dominantly argued against Ni isotope anomalies in bulk meteorites (Figs. 1–3), as did Chen et al. (2009) for their multi-collector thermal ionization mass-spectrometry (MC–TIMS) analyses (Fig. 3). As exceptions to these overall observations of bulk Ni isotope homogeneity, Quitté et al. (2006) reported correlated, negative values of ε60Ni61/58 and ε62Ni61/58 in many of the sulfide inclusions they analyzed from iron meteorites (Fig. 2a). Subsequently, Cook et al. (2008) reported more modest anomalies in troilites from iron meteorites (Fig. 2). Both studies argued for the preservation of a presolar component in these sulfide inclusions, although how this occurred mechanistically was problematic. The MC–TIMS work of Chen et al. (2009) provided a different measurement perspective. This study argued against resolvable differences in ε60Ni61/58 and ε62Ni61/58 in either bulk or sulfide samples, at a level of ±0.2ε and ±0.5ε respectively (Fig. 2a). Since then, no one has further pleaded for the case of anomalous sulfides and the original analyses seem likely to have been measurement artefacts. However, there has been on-going debate about the presence of Ni isotopic anomalies in bulk meteorites. In striking contrast to the bulk meteorite analyses described above, Bizzarro et al. (2007) reported a dataset with near constant negative ε60Ni61/58 (and ε62Ni61/58) in differentiated meteorites but ε60Ni61/58 ~0 and positive ε62Ni61/58 in chondrites. These data were used to invoke a late super-nova injection of 60Fe into the solar system. Subsequent studies were unable to reproduce these results (Dauphas et al. 2008; Regelous et al. 2008; Chen et al. 2009; Steele et al. 2011; Tang and Dauphas 2012, 2014) and noted that the systematics of the Bizzarro et al. (2007) dataset were consistent with an interference on 61Ni. In reporting the results of further analyses, Bizzarro et al. (2010) commented that their new data were inconsistent with Bizzarro et al. (2007) but agreed with the observations of Regelous et al. (2008). The data from Bizzarro et al. (2007) will thus not be further considered. Regelous et al. (2008) presented bulk analyses of ε60Ni61/58 and ε62Ni61/58 on a suite of chondrites and iron meteorites with precisions of around ± 0.02ε and ± 0.04ε respectively. By making higher precision measurements, in part by pooling multiple repeats of the same sample and by examining a wider range of meteorites than earlier studies, Regelous et al. (2008) were able to resolve differences in bulk meteorite compositions (Fig. 3a). They illustrated that variablity between different chondrite groups is largely echoed by that in iron meteorites (Fig. 3a). Notably the IVB irons have Ni isotopic compositions similar to carbonaceous chondrites (positive ε62Ni61/58), whilst the other magmatic irons resemble ordinary chondrites (with negative ε62Ni61/58). As for a number of other isotopic systems, enstatite chondrites were largely within error of terrestrial values. These observations were further refined at higher precision (Fig. 4a) and with the inclusion of ε64Ni61/58 data (Fig. 4b) by Steele et al. (2011, 2012) and Tang and Dauphas (2012, 2014). These data revealed a continuous, well defined array in ε62Ni61/58 vs ε64Ni61/58 from ordinary chondrites and most magmatic irons, through terrestrial values in EH chondrites to carbonaceous chondrites and IVB irons (Fig. 4b). This ordering of meteorite groups is the same as observed in the mass-independent isotopic compositions of other first row, transition elements (Trinquier

The Isotope Geochemistry of Ni

Cook et al. Quitte et al. Chen et al.

0

ε60Ni58/61

517

−5

−10

−15

(a) −20

−25

−20

−15

−10

−5

0

ε62Ni58/61

5

4 2

ε64Ni58/61

0 −2 −4 −6 −8 −10 −12 −3

(b) −2

−1

0

1

ε62Ni58/61

2

3

4

Figure 2. Mass independent nickel isotope data of sulfide inclusions from iron meteorites (Quitté et al. 2006; Cook et al. 2008; Chen et al. 2009), (a) ε60Ni58/61 vs. ε62Ni58/61 and (b) ε64Ni58/61 vs. ε62Ni58/61. Again the dashed lines indicate trajectories caused by error or interference (moving to the lower left) on 61Ni. It appears that 61Ni error may be a dominant (but not sole) cause of variability in the highly anomalous sulfides. The blue boxes indicate the dimensions of parameter space shown in Figure 4.

et al. 2007, 2009). Although the total isotopic variability in Ni is smaller than for Ti or Cr, its notable strength it that both chondrites and iron meteorites can be analyzed to high precision, allowing genetic associations to be made from bulk compositions of iron meteorites rather than from occasional oxygen-bearing inclusions they contain (e.g. Clayton et al. 1983). Strikingly, the bulk Ni isotope array in Figure 4b points towards the CAI values presented by Birck and Lugmair (2008), see Figure 1b. For the same arguments as made by Trinquier et al. (2009), however, the bulk meteorite array is not formed by simple mixing between a single bulk composition and CAI, which would created a strongly curved array (see Fig. 5). Carbonaceous chondrites are enriched, relative to ordinary chondrites, in the same isotopic component that is manifest more strongly in CAIs. Yet, it is not addition of CAIs themselves that causes the trend in Figure 4b but presumably variable abundances of specific pre-solar grains in chondrite matrices (see Dauphas et al. 2010 and Qin et al. 2011 for the Cr isotope case).

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ε60Ni58/61

0.4 0.2

Carbonaceous Chondrites Ordinary Chondrites Enstatite Chondrites Iron meteorites Dauphas et al. Regelous et al. Chen et al.

0.0

−0.2 −0.4 −0.6

(a) −0.6 −0.4 −0.2

0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.6

ε62Ni58/61

ε64Ni58/61

0.8

0.4

0.0

−0.4

(b) −0.6 −0.4 −0.2

ε62Ni58/61

Figure 3. Mass independent nickel isotope data a) ε60Ni58/61 vs. ε62Ni58/61 and b) ε64Ni58/61 vs. ε62Ni58/61 of bulk meteorite analyses from ‘second generation’ studies (Dauphas et al. 2008; Regelous et al. 2008; Chen et al. 2009). The samples of Regelous et al. (2008) show small but resolved anomalies in both chondritic and iron meteorites. These analyses are consistent with the less precise data of Dauphas et al. (2008), who conversely argued against bulk Ni isotopic variability (as did Chen et al. 2009). Dashed lines represent 61Ni error vectors and solid line with grey band the best fit array of Steele et al. (2012), as discussed in caption to Figure 1. Again (dashed) blue boxes indicate the dimensions of parameter space shown in Figure 4.

The well-defined array in non-radiogenic isotopes (Fig. 4b) provides key constraints on the nucleosynthetic origins of this important nebular component. At face value it represents coupled enrichments in the neutron rich isotopes of 62Ni and 64Ni. However, the 3:1 slope of the array (Fig. 4b) can also be reproduced by variable meteoritic values of 58Ni/61Ni, the normalizing isotope ratio. Indeed, from high-precision mass-dependent isotopic measurements (see the section Mass-Dependent Ni isotopic Variability), Steele et al. (2012) showed that variability in 58Ni best explains all observations. This implies the source of this anomalous Ni is from the Si–S zone of a SNII. This contrasts with material from the O–Ne zone required to account for a similarly constrained component in the Ti isotopic system. Steele et al. (2012) suggested ways in which the different, contributing zones for these different elements might be reconciled by grains from different zones being sorted (homogenized or unmixed) by solar system processes. However, these issues remain unresolved.

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ε60Ni58/61

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Carbonaceous Chondrites Ordinary Chondrites Enstatite Chondrites Iron meteorites Steele et al. Tang and Dauphas. Earth

0.2 0.0

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(b) −0.2

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ε62Ni58/61

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Figure 4. Mass independent Ni isotope measurements a) ε60Ni58/61 vs. ε62Ni58/61 and b) ε64Ni58/61 vs. ε62Ni58/61 on bulk meteorites from the highest precision studies (Steele et al. 2011, 2012; Tang and Dauphas 2012, 2014). Data from these studies show a consistent set of anomalies between carbonaceous, ordinary and enstatite chondrites and different groups of iron meteorites. There is a strong positive correlation between ε62Ni58/61 and ε64Ni58/61, first identified by Steele et al. (2012) and confirmed by Tang and Daphaus (2012). This is array quite distinct from the dashed 61Ni error line. A least squares York regression and the 2σ uncertainty error envelope (York 1969; Mahon 1996) through the bulk meteorite and peridotite data of Steele et al. (2012) is shown with black line and grey band and yields a slope of 3.003 ± 0.116. Due to the bias in least squares regressions when using averages of repeat sample measurements and individual uncertainties, Steele et al. (2012) used each individual measurement of each sample and the homoscedastic uncertainty (see Appendix A.2 of Steele et al. 2012 for a discussion). Since individual measurement of meteorite samples are not available from Tang and Dauphas (2012, 2014) these data have not been included in the regression, but there is clearly excellent agreement between all three datasets.

The correlation of bulk analyses of ε60Ni61/58 with the other isotope ratios (ε62Ni61/58 or ε Ni61/58) is much less systematic (Regelous et al. 2008; Steele et al. 2012; Tang and Dauphas 2014), see Figure 4a. In general, carbonaceous chondrites have lower ε60Ni61/58 than ordinary chondrites and enstatite chondrites, but CI chondrites are a notable exception. It is tempting to attribute these different relationships to the radiogenic nature of ε60Ni61/58, but given the abundance of 60Fe inferred from bulk meteorite studies (see the section Extinct 60Fe and radiogenic 60Ni below) this seems unlikely. Instead, this presumably reflects part of a more complex nucleosythetic 64

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4% 3% 2%

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Calcium, aluminium rich inclusions Carbonaceous Chondrites Ordinary Chondrites Enstatite Chondrites Earth

0 −2 −0.5 0.0

0.5

1.0

64

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ε Ni58/61

2.5

3.0

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Figure 5. Plot of ε Ni58/61 vs ε Ti47/49 for bulk meteorites (Ti isotope data from Trinquier et al. (2009), Ni data from Birck and Lugmair (1988) and Steele et al. (2012)). A linear array is defined by analyses of ordinary, enstatite and CI chondrites and this extends to intersect with CAI compositions (see caption to Fig. 4). However, CAI-rich meteorite groups (CV CO) plot above the array, likely reflecting analyses of unrepresentative sub-samples of these meteorites with an excess of CAI ~ 3–4% by mass (illustrated with a mixing curve from a CI composition), see Hezel et al. (2008). The high Ti/Ni of CAIs means mixing trajectories with bulk chondrite compositions are highly convex up. Thus simple mixing between CAIs and ordinary chondrites (black curve) cannot explain the straight bulk meteorite array. 64

50

signature, with several components, potentially one which incorporates high fossil 60Fe, involved in generating the Ni isotopic compositions of ordinary and carbonaceous chondrites. The carriers of the exotic Ni isotopic components that shape variable bulk meteorite compositions remain to be identified. The highly anomalous isotopic compositions found in separated SiC grains from the CM2 meteorite, Murchison (Marhas et al. 2008) do not readily account for the mass-independent Ni isotope variations seen in bulk samples. These ion-probe analyses show 61Ni/58Ni and 62Ni/58Ni ratios > 1000ε higher than terrestrial values (for these extreme ratios there is no internal normalization) in X-grains believed to be derived from SNII sources. These Ni isotopic signatures suggest derivation from outer He/N and He/C zones of an SNII event, different to those inferred by Steele et al. (2012) to be necessary to account for bulk isotopic variability. As with other analyses of pre-solar SiC, these measurements bear striking testimony to the diversity of stellar sources that contribute to the bulk composition of the solar system, but do not identify a carrier for the signature that causes variability between different bulk, meteoritic objects (Steele et al. 2012).

EXTINCT 60Fe AND RADIOGENIC 60Ni As alluded to in the preceding section, much of the initial interest in Ni isotope cosmochemistry was focussed on trying to identify the presence of live 60Fe in the early solar system. A salient property of 60Fe is that it is only made during stellar nucleosynthesis

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and cannot be produced by particle irradiation in our own solar system, unlike 26Al. Thus an initial solar system value of 60Fe/56Fe (60Fe/56Fe°SSI) significantly higher than the calculated steady state background of the interstellar medium (ISM), recently estimated at ~3 × 10 − 7 by Tang and Dauphas (2012), would provide strong evidence for injection of short-lived nuclides by a proximal stellar explosion. Such an explanation, linked to a trigger for solar system formation, has long been invoked to account for the elevated initial solar system 26 Al/27Al (e.g. Cameron and Turan 1977), but alternative means of generating 26Al (e.g. Clayton and Jin 1995) have made a more definitive test using 60Fe highly desirable (see Wasserburg et al. 1998). As explored below, values of 60Fe/56Fe°SSI determined from Ni isotopic measurements of meteorites remain contentious although our own perspective is that values above those of the ISM are questionable. In a pioneering contribution, Shukolyukov and Lugmair (1993a) reported evidence for live 60Fe from analyses of 60Ni/58Ni on samples of the eucrite Chervony Kut. Eucrites represent a good target for detecting in situ decay of 60Fe, given their extremely high Fe/Ni (>10000). Such high Fe/Ni ratios are a consequence of core formation at low pressures (i.e. on a small planetary body) and subsequent magmatic fractionation. Given the formation and differentiation of the eucrite parent body within the first few million years of solar system history (e.g. Papanastassiou and Wasserburg 1969) the resultant Fe/Ni were likely to generate resolvable 60Ni anomalies given significant initial 60Fe. Different components of Chervony Kut showed elevated ε60Ni62/58 (up to 50) and three bulk samples with a range of Fe/Ni defined a straight line with a slope that implied 60Fe/56Fe° ~ 4 × 10 − 9 (Fig. 6a). The authors used an age of 10 ± 2 Ma post CAI, from the similarity of the meteorites’ 87Sr/86Sr to dated angrites (Lugmair and Galer 1992), to infer 60Fe/56Fe°SSI ~ 2 × 10 − 6. Since then, determinations of the timing of differentiation of the eucrite parent body give older ages, ~3Ma post CAI (Lugmair and Shukolyukov 1998; Bizzarro et al. 2005; Trinquier et al. 2008; Schiller et al. 2011) and the accepted half-life of 60Fe has increased from 1.49 Ma (Kutschera et al. 1984) to 2.62 Ma (Rugel et al. 2009). As a result of these changes a more contemporary interpretation of these results would yield 60Fe/56Fe°SSI ~ 2 × 10− 8. Shukolyukov and Lugmair (1993a) originally noted that their value of 60Fe/56Fe°SSI was in keeping with ε60Ni61/58 measurements made on CAIs by Birck and Lugmair (1988), assuming that such 60Ni excesses were radiogenic. From the discussion of nucleosynthetic variability in the section Nucleosynthetic Ni isotopic variations, it should be clear that this assumption is by no means valid, nor is the value thus derived consistent with the revised 60Fe/56Fe°SSI. The magnitude of possibly radiogenic ε60Ni61/58 in CAIs is no bigger than ε62Ni61/58, which must be nucleosynthetic (Fig. 1a). Although Kruijer et al. (2014) demonstrated that it is possible to disentangle nucleosynthetic from radiogenic contributions to 182W/184W in CAIs in order to define 182Hf/180Hf°SSI, neither the abundance of initial parent nor parent–daughter fractionation in CAIs are sufficiently large to make this approach viable for determining 60Fe/56Fe°SSI. Indeed, most CAIs have sub-chondritic 56Fe/58Ni (e.g. Quitté et al. 2007), as a result of the slightly more refractory cosmochemical behavior of Ni relative to Fe. Thus at best, minor ε60Ni61/58 deficits in bulk CAI would be expected if 60Fe/56Fe°SSI were large enough for the bulk nebula to evolve to more radiogenic ε60Ni61/58, as is the case for the 53Mn–53Cr system (Birck and Allègre 1985). Moreover, internal CAI isochrons offer little scope for constraining 60 Fe/56Fe°SSI given both modest Fe/Ni fractionation between common phases in CAIs and the frequent disturbance of Fe/Ni ratios in CAIs specifically (Quitté et al. 2007) and chondritic meteorites in general (e.g. Telus et al. 2016). In contrast to the 26Al–26Mg system, but similar to the 53Mn–53Cr pair, bulk CAI measurements therefore offer scant opportunity to determine the initial parent abundance of the 60Fe–60Ni system. Instead, basaltic achondrites, with their highly fractionated Fe–Ni, have been the focus of much additional work to constrain 60Fe/56Fe°SSI.

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Figure 6. Representative isochron diagrams for the 60Fe–60Ni system measured by different analytical techniques. (a) TIMS isochron of different bulk samples of the eucrite Chervony Kut (Shukolyukov and Lugmair 1993a). (b) MC–ICPMS isochron of mineral separates from the quenched angrite D’Orbigny (Tang and Dauphas 2012). (c) SIMS isochron of a single Efremovka chondrule (Ch 1), Mishra and Chaussidon (2014). Also shown are recalculated least squares linear York regressions (York 1968, Mahon 1995) for each dataset, yielding 60Fe/56Fe°. Note the highly contrasting values obtained for 60Fe/56Fe° by bulk analyses in (a) and (b) versus in situ analyses (c) of objects of comparable age.

In a follow-on study, Shukolyukov and Lugmair (1993b) analyzed the eucrite Juvinas and noted an order of magnitude lower 60Fe/56Fe° (~4 × 10 − 10), which the authors attributed to ~4 Ma evolution of the eucrite parent body mantle source between generation of the basaltic melts represented by Chervony Kut and Juvinas. Using the newer 60Fe half-life this period would be ~9 Ma, a value which resonates with the age difference of ~11 Ma inferred from Hf–W analyses between a group of eucrites including Juvinas and an older group of

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eucrites (Touboul et al. 2015); sadly Chervony Kut itself was not analyzed for its Hf–W systematics. As with Chervony Kut (Shukolyukov and Lugmair 1993a), separated mineral phases from Juvinas did not show meaningful isochronous relations and Shukolyukov and Lugmair (1993b) argue that the mobility/diffusivity of Ni made it susceptible to resetting on this shorter length scale, especially given the complex history of eucrites, with well documented thermal metamorphism (e.g. Takeda and Graham 1991). Subsequent studies of eucrites have been made with increasingly precise analyses using MC–ICPMS, but in essence show comparable features. Quitté et al. (2011) reported data for Bouvante and further analyses of Juvinas. Whole rock sub-samples of Bouvante define two arrays and if these are taken to be isochrons they imply 60Fe/56Fe° = 5 × 10 − 9 (but with an implausibly negative intercept of ε60Ni62/58 = −24 ± 3) and 60Fe/56Fe° = 5 × 10 − 10 (with a reasonable, near zero ε60Ni62/58 intercept). Instead, the authors argue that the data represent two mixing lines between clasts of different compositions. Whilst the second array gives a similar value of 60Fe/56Fe° to Juvinas, as obtained by Shukolyukov and Lugmair (1993b), additional measurements of Juvinas by Quitté et al. (2011) are inconsistent with the array of Shukolyukov and Lugmair (1993b). Quitté et al. (2011) obtain a higher 60Fe/56Fe° (2 × 10 − 9), using their two unwashed bulk samples and two unwashed, lower Fe/Ni samples from the Shukolyukov and Lugmair (1993b) array (ignoring the washed samples which they argue may have suffered Fe–Ni fractionation as a result of this preparation). As with the work of Shukolyukov and Lugmair (1993b), the Juvinas mineral analyses of Quitté et al. (2011) are scattered and presumably perturbed by secondary processes. There is clear difficulty in distinguishing primary from secondary signatures in these brecciated, thermally metamorphosed meteorites, but none of the eucrite arrays define 60Fe/56Fe° greater than 5 × 10 − 9. Tang and Dauphas (2012) comprehensively reassessed this issue using a collection of bulk eucrites and diogenites. This dataset gave a large range in Fe/Ni, which coupled with their high precision measurements (typical ε60Ni61/58 better than ± 0.3), provided a more definitive 60 Fe/56Fe° = (3.5 ± 0.3) × 10 − 9 for the eucrite parent body. This result is notably in keeping with the higher values obtained from ‘internal isochrons’ on sub-samples of meteorites discussed above (e.g. Fig. 6a). Using an age of 2.4 ± 1.1 Ma post CAI for silicate differentiation of 4 Vesta (Trinquier et al. 2008; Connelly et al. 2012), which presumably sets the variable Fe/Ni seen in the eucrites, these data yield a 60Fe/56Fe°SSI (6.6 ± 2.5) × 10 − 9. This work of Tang and Dauphas (2012) also yields a bound of 4 ± 2 Ma on the age of core formation on 4 Vesta, from a two-stage evolution model of its mantle. Namely, the ε60Ni61/58 of the mantle, determined by the intercept of the eucrite–diogenite array with an estimated bulk mantle 56Fe/58Ni ~ 2700 informs on the time since core formation. Tang and Dauphas (2014) subsequently used a similar approach to constrain the timing of core formation and hence growth of Mars. They argued that the planet reached 44% of its size no earlier than 1.2 Ma post CAI or otherwise the ε60Ni61/58 of the SNC meteorites they measured would be more radiogenic. Quenched angrites provide a more petrologically robust sample for determining 60Fe/56Fe°, even if their Fe/Ni are not quite as extreme as the eucrites (cf. Figs. 6a,b). Moreover, the welldefined ages for these samples determined using extant isotope chronometry (Amelin 2008a,b; Connelly et al. 2008, Brennecka and Wadhwa 2012), potentially provide more accurate decay correction in calculating 60Fe/56Fe°SSI. Three independent studies (Quitté et al. 2010; SpivakBirndorf et al. 2011; Tang and Dauphas 2012) obtained consistent values for internal isochrons of d’Orbigny (e.g. Fig. 6b) which give a weighted average 60Fe/56Fe° = (3.3 ± 0.5) × 10 − 9. Two internal isochrons from a second quenched angrite, Sahara 99555, are also in mutual agreement but yield a lower weighted mean 60Fe/56Fe° = (1.9 ± 0.4) × 10 − 9 (Quitté et al. 2010; Tang and Dauphas 2015). Tang and Dauphas (2015) convincingly argue this difference relative to d’Orbigny likely reflects terrestrial weathering experienced by Sahara 99555.

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In all, these angrite 60Fe/56Fe° are in keeping with those of eucrites, suggesting a similar initial Fe and timing of planetary differentiation on these two bodies. So a reassuringly consistent value has emerged from these various TIMS/MC–ICPMS studies which span several decades of work. The data from d’Orbigny provides the best constrained value and using an age of 3.9 Ma post CAI (Amelin 2008a; Brennecka and Wadhwa 2012; Connelly et al. 2012) we calculate 60 Fe/56Fe°SSI =(9.8 ± 4.5) × 10 − 9. We note this value is lower than the equivalent cited by Tang and Dauphas (2012) as a consequence of our using the Pb–Pb ages rather than Mn–Cr chronometry. 60

Further bulk analyses of a range of meteoritic materials are supportive of the low Fe/56Fe°SSI determined from angrite and eucrite analyses, albeit from less well constrained scenarios. Shukolyukov and Lugmair (1993b) and Quitté et al. (2010) reported no systematic differences in ε60Ni62/58 for various bulk samples (ureilites) and separated phases (e.g. troilite) with high but variable Fe/Ni ratios. Moynier et al. (2011) placed a maximum upper bound on 60Fe/56Fe°SSI of 3 × 10 − 9 from the absence of 60Ni isotope anomalies in measurements of troilite from the iron meteorite Muonionalusta. These troilites have Pb–Pb model ages as old as the quenched angrites (Blichert-Toft et al. 2010), which coupled with their high Fe/Ni (up to 1500) should result in radiogenic ε60Ni61/58 given sufficiently high 60Fe/56Fe°. Although appealing targets for analysis, re-equilibration of the troilites with the surrounding Ni-rich metal during parent body during cooling would tend to erase any ε60Ni61/58 anomalies. The authors briefly argue against such an interpretation on the basis of the preservation of ancient Pb–Pb ages, but the potential for diffusional exchange of Ni with the Ni-rich host metal (see Chernonozhkin et al. 2016) seems much greater than for Pb. Whilst the conclusions of Moynier et al. (2011) are thus compatible with other studies, whether or not the measurements represent an independent constraint on 60Fe/56Fe°SSI remains open to debate. 60

Analysis of chondrules from the CBa meteorite Gujba and ungrouped 3.05 ordinary chondrite NWA 5717 by Tang and Dauphas (2012) form near horizontal arrays that yield 60Fe/56Fe° from 1 to 3 × 10 − 9. However, elemental mapping by Telus et al. (2016) showed that chondrules in all chondrites they studied had experienced some open system behavior of Fe and Ni. Only the most pristine, LL3.0 meteorite, Semarkona, retained undisturbed chondrules, about ~40% of those studied. Prompted by these findings, Tang and Dauphas (2015) made measurements of single chondrules from Semarkona, to yield a valuable but still relatively poorly defined 60 Fe/56Fe° = (5 ± 3) × 10 − 9. The lack of significant differences between 60Fe/56Fe° for these meteorites of different metamorphic grade suggests that they are not unduly compromised by this open system behavior. Given an average chondrule age of 2Ma post CAI (see recent compilation of data in Budde et al. 2016), the 60Fe/56Fe°SSI ~9 × 10 − 9 derived from these individual chondrule measurements is notably compatible with the studies from achondrite meteorites. In a grand compilation of various determinations of 60Fe/56Fe° from bulk measurements, Tang and Dauphas (2015) derived a weighted average 60Fe/56Fe°SSI = (1.0 ± 0.3) × 10 − 8. In contrast to the work described above on Ni separated from bulk samples, much higher 60Fe/56Fe° have been inferred from in situ work by SIMS. Initially, the absence of detectable differences in the 60Ni/61Ni of olivines from type II chondrules from Semarkona, relative to terrestrial olivines, was used to place an upper limit of 3.4 × 10 − 7 on their 60Fe/56Fe° (Kita et al. 2000). However, later studies documented correlated ε60Ni and Fe/Ni in matrix sulfides and oxides from primitive ordinary chondrites suggesting 60Fe/56Fe° from 1 × 10 − 7 to 1 × 10 − 6 (Tachibana and Huss 2003; Mostefaoui et al. 2004, 2005; Guan et al. 2007). As the time of formation of these phases is uncertain, the significance of these data arrays for inferring 60Fe/56Fe°SSI was open to question. In an elegant study, Tachibana et al. (2006) subsequently analysed different phases from the chondrules of Semarkona. Although the ranges in correlated Fe/Ni and ε60Ni62/61 were lower than in the sulfide work, interpretation

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of the arrays as constraining 60Fe/56Fe°SSI (5–10) × 10 − 7 seemed less equivocal. Similar results were reported by Mishra et al. (2010) for single chondrule analyses from a wider range of unequilibrated ordinary chondrites. Yet, all such analyses are controlled by the errors on the very small 61Ni and 62Ni beams used for determining ε60Ni62/61 (the larger 58Ni beam cannot be used as it is interfered by 58Fe). These measurements thus critically require accurate background determination and interference free spectra. Moreover, the use of the minor Ni isotope in the denominator of such low intensity measurements can lead to a statistical bias in calculated ratios (Ogliore et al. 2011, see also Coath et al. 2013). This artefact resulted in spurious correlations between Fe/Ni and ε60Ni62/61 in all earlier work (Telus et al. 2012). Yet subsequent work has continued to report high inferred 60Fe/56Fe° (see Fig. 6c) from in situ analyses of chondrules in studies for which such statistical bias is argued to be insignificant (Mishra and Chaussidon 2014; Mishra and Goswami 2014; Mishra et al. 2016). Hence, inferred 60Fe/56Fe° from TIMS/MC–ICPMS studies and SIMS analyses of individual chondrules are markedly different. This contrast in conclusions from bulk and in situ approaches also extends to other MC–ICPMS work. Regelous et al. (2008) and Steele et al. (2012) obtained the loose constraint that the 60Fe/56Fe° of carbonaceous chondrites was